1. "Catholics teach that Christ is “physically present” in the Mass."
Incidentally, there is a misnomer in that sentence, namely the misnomer of referring those in the Roman communion as "Catholics." The Roman church is not the universal (that's what "Catholic" means) church of Christ. But that's not what Mr. Castaldo has in mind. Mr. Castaldo actually tries to argue that Christ not physically present "in the Mass."
There is not a teaching that Christ is present physically at the start of the Mass, but it is accurate to say that "physical presence" is the Roman teaching (though it is not the whole of the teaching). For example:
A third element, that has an increasingly natural and central place in World Youth Days and in the spirituality that arises from them, is adoration. I still look back to that unforgettable moment during my visit to the United Kingdom, when tens of thousands of predominantly young people in Hyde Park responded in eloquent silence to the Lord’s sacramental presence, in adoration. The same thing happened again on a smaller scale in Zagreb and then again in Madrid, after the thunderstorm which almost ruined the whole night vigil through the failure of the microphones. God is indeed ever-present. But again, the physical presence of the risen Christ is something different, something new. The risen Lord enters into our midst. And then we can do no other than say, with Saint Thomas: my Lord and my God! Adoration is primarily an act of faith – the act of faith as such. God is not just some possible or impossible hypothesis concerning the origin of all things. He is present. And if he is present, then I bow down before him. Then my intellect and will and heart open up towards him and from him. In the risen Christ, the incarnate God is present, who suffered for us because he loves us. We enter this certainty of God’s tangible love for us with love in our own hearts. This is adoration, and this then determines my life. Only thus can I celebrate the Eucharist correctly and receive the body of the Lord rightly.Christmas Greetings to the Roman Curia, 22 December 2011 (link - video link)(emphasis added)
Notice that Benedict XVI (who is not just the current pope, but also a theologian within his church) treats the sacramental presence as a physical presence, and therefore distinguishable from the spiritual omnipresence of God.
Moreover, Benedict XVI's view is not a mistake (or a "misnomer" if you prefer). As CCC 1373 explains: "he is present . . . most especially in the Eucharistic species." CCC 1374 goes into more detail (emphasis added):
The mode of Christ's presence under the Eucharistic species is unique. It raises the Eucharist above all the sacraments as "the perfection of the spiritual life and the end to which all the sacraments tend." In the most blessed sacrament of the Eucharist "the body and blood, together with the soul and divinity, of our Lord Jesus Christ and, therefore, the whole Christ is truly, really, and substantially contained." "This presence is called 'real' - by which is not intended to exclude the other types of presence as if they could not be 'real' too, but because it is presence in the fullest sense: that is to say, it is a substantial presence by which Christ, God and man, makes himself wholly and entirely present."
Moreover, this presence is unique because it is bodily (i.e. physical) presence: "It is by the conversion of the bread and wine into Christ's body and blood that Christ becomes present in this sacrament." (CCC 1375) The presence of Christ is not a visible presence: "Since Christ was about to take his departure from his own in his visible form, he wanted to give us his sacramental presence." (CCC 1380)
Mr. Castaldo makes an argument:
When describing Jesus Christ as the Eucharist, Catholics will say that the Lord is “really,” “truly,” “wholly,” “continuously,” or “substantially” present, but not "physically.” To state the Jesus is “physically” present is to suggest that he is present “locally” (as he is now in heaven at the right hand of the Father). The Eucharistic presence of Christ, although understood as no less real, is sacramentally present in the transubstantiated host. From the Catechism of the Catholic Church:The positive aspects of his comments are of course right: Rome teaches that Jesus is “really,” “truly,” “wholly,” “continuously,” and “substantially” present. But Mr. Castaldo wrongly reasons from the fact that "physical" is not used, to suppose that "physical presence" is denied. We have observed Benedict XVI using such an expression - but consider further: before the consecration, there is just bread and wine. After the consecration, there is no more bread and wine. What appears to be bread and wine according to all of science and reason is, Rome claims, the body, blood, soul, and divinity of Christ.
1413 By the consecration the transubstantiation of the bread and wine into the Body and Blood of Christ is brought about. Under the consecrated species of bread and wine Christ himself, living and glorious, is present in a true, real, and substantial manner: his Body and his Blood, with his soul and his divinity (cf. Council of Trent: DS 1640; 1651).
Moreover, that presence is "local" in the sense of being contained. As CCC 1367 explains: "in the Mass, the same Christ who offered himself once in a bloody manner on the altar of the cross is contained ... ." That's why the storage container for the consecrated hosts is called a "tabernacle." It is because it provides a housing for what Rome falsely claims is Jesus himself.
Mr. Castaldo's denial of the physical and local presence of Christ seems to run contrary to the teachings of Pope Paul VI:
This presence is called "real" not to exclude the idea that the others are "real" too, but rather to indicate presence par excellence, because it is substantial and through it Christ becomes present whole and entire, God and man. And so it would be wrong for anyone to try to explain this manner of presence by dreaming up a so-called "pneumatic" nature of the glorious body of Christ that would be present everywhere; or for anyone to limit it to symbolism, as if this most sacred Sacrament were to consist in nothing more than an efficacious sign "of the spiritual presence of Christ and of His intimate union with the faithful, the members of His Mystical Body."(Paul VI, Mysterium Fidei, 3 September 1965, section 39)
But I suspect that Mr. Castaldo's argument comes (directly or indirectly) from Thomas Aquinas who himself took the position that Christ is not present "locally." But by that he did not deny that Christ's presence is physical, as you can see:
... Christ's body is in this sacrament not after the proper manner of dimensive quantity, but rather after the manner of substance. But every body occupying a place is in the place according to the manner of dimensive quantity, namely, inasmuch as it is commensurate with the place according to its dimensive quantity. Hence it remains that Christ's body is not in this sacrament as in a place, but after the manner of substance, that is to say, in that way in which substance is contained by dimensions; because the substance of Christ's body succeeds the substance of bread in this sacrament: hence as the substance of bread was not locally under its dimensions, but after the manner of substance, so neither is the substance of Christ's body. Nevertheless the substance of Christ's body is not the subject of those dimensions, as was the substance of the bread: and therefore the substance of the bread was there locally by reason of its dimensions, because it was compared with that place through the medium of its own dimensions; but the substance of Christ's body is compared with that place through the medium of foreign dimensions, so that, on the contrary, the proper dimensions of Christ's body are compared with that place through the medium of substance; which is contrary to the notion of a located body.(Summa Theologica, 3a, 76, 6)
Hence in no way is Christ's body locally in this sacrament.
Reply to Objection 1. Christ's body is not in this sacrament definitively, because then it would be only on the particular altar where this sacrament is performed: whereas it is in heaven under its own species, and on many other altars under the sacramental species. Likewise it is evident that it is not in this sacrament circumscriptively, because it is not there according to the commensuration of its own quantity, as stated above. But that it is not outside the superficies of the sacrament, nor on any other part of the altar, is due not to its being there definitively or circumscriptively, but to its being there by consecration and conversion of the bread and wine, as stated above (1; 15, 2, sqq.).
Reply to Objection 2. The place in which Christ's body is, is not empty; nor yet is it properly filled with the substance of Christ's body, which is not there locally, as stated above; but it is filled with the sacramental species, which have to fill the place either because of the nature of dimensions, or at least miraculously, as they also subsist miraculously after the fashion of substance.
Reply to Objection 3. As stated above (Article 4), the accidents of Christ's body are in this sacrament by real concomitance. And therefore those accidents of Christ's body which are intrinsic to it are in this sacrament. But to be in a place is an accident when compared with the extrinsic container. And therefore it is not necessary for Christ to be in this sacrament as in a place.
Of course, Thomas' views on this (see the rest of them) are not de fide for those in the Roman Catholics, but certainly are influential. It's not clear to me that Mr. Castaldo understands what Aquinas is saying about the body of Christ not being locally present, but to deny that the body is not physically present is not only inconsistent with Benedict XVI (as mentioned above) and rationally with the de fide pronouncements of Trent but also inconsistent with Aquinas himself:
Objection 2. Further, the form of Christ's body is His soul: for it is said in De Anima ii, that the soul "is the act of a physical body which has life in potentiality". But it cannot be said that the substantial form of the bread is changed into the soul. Therefore it appears that it remains after the consecration.(Summa Theologica, 3a, 75, 6)
Reply to Objection 2. The soul is the form of the body, giving it the whole order of perfect being, i.e. being, corporeal being, and animated being, and so on. Therefore the form of the bread is changed into the form of Christ's body, according as the latter gives corporeal being, but not according as it bestows animated being.
The next alleged error is that "[Roman] Catholics teach that Christ is re-sacrificed at the Mass." That accurately reflects the bizarre contemporary teaching that there is only one sacrifice and yet every mass is a sacrifice. But in the discussion of Purgatory, the Council of Trent did not hesitate to speak of the "sacrifices of the masses" ("missarum ... sacrificia") as being of assistance to those in Purgatory.
But whether or not Rome today maintains Trent or contradicts it, we may still object that the Mass amounts to a new sacrifice, inasmuch as the Mass purports to "re-present" (not represent) and perpetuate the sacrifice that took place on Calvary.
3. Multiple Deaths?
The final alleged mistake was: "[Roman] Catholics teach that Christ dies at the Mass." It certainly is not (to my knowledge) de fide that Christ dies at each mass. However, that in itself is problematic for the person in the Roman communion. How can there be a sacrifice of the victim without the death of the victim? But that's part of the objection, not part of the teaching to which we are responding, technically.