Here is a summary of the requested Se'if with some of the more important notes of the Mishnah Berurah (which I have referred to as M.B.).

'One is not permitted to cook on behalf of Nochrim on Yom-Tov. Consequently, one is forbidden to invite him for a meal on Yom-Tov, since one is then liable to cook an extra dish in his honor. One may however, send food to his house (using food that he has cooked for himself [M.B.], though in certain circumstances, some Poskim permit it) via a Nochri Sheli'ach (assuming that it is forbidden to carry the food to the Nochri's house [see Ba'er Heitev]).

One is also allowed to invite his Nochri servant or maidservant, a Nochri Sheli'ach whom someone sent to him and a Nochri who came of his own accord (unless he is an important person, when it may be forbidden [see M.B.]). In all these cases, we are not afraid that one may come to cook extra on behalf of his guest.

When it comes to his Nochri servant and maidservant, one is even permitted to add to the pot in which he is cooking his own food (though he is not permitted to tell them this specifically M.B., and even then, it may not be permitted if he adds food to the pot separately [see M.B.]), but not for any of the other above Nochrim.

If a Yisrael is cooking in an oven belonging to a Nochri, and is obligated to give one loaf to the owner, he should take care not to designate that loaf before baking, since he will then be baking the loaf of a Nochri. What he must do is to first bake all the loaves that he needs to bake and designate the Nochri's loaf afterwards.

be'Virchas Kol Tuv

Eliezer Chrysler