Tefillin: The rule is that one can do a mitzva on someone else’s behalf only if the person doing it is himself obligated in the mitzva. Obviously, a non-Jew is not obligated to put on tefillin. Thus, the matter depends on the following chakira (analytical dilemma): Is the mitzva the action of putting on and fastening the tefillin, in which case, only one who is obligated in tefillin may perform it? Alternatively, is the mitzva the state of having them on the arm and head, in which case, it is not important who places them on the person? The gemara‎{&&1}‎ tells of a woman (who is not obligated in tefillin) who would place tefillin on her husband. The Har Tzvi‎{&&2}‎ derives from this that the mitzva is the state of wearing tefillin; the action is just preparatory. Therefore, the non-Jew can put the tefillin on the Jew. The Jew will make the beracha.
Cooking:‎{&&3}‎ Regarding the issue of bishul akum (cooking done by a non-Jew), one can solve the problem as follows: Regarding bread baked by a non-Jew, it is sufficient for a Jew to light the flame from which the flame for baking is taken. There is a critical machloket if this leniency applies to all foods or whether a Jew must take part in the actual cooking. The Shulchan Aruch‎{&&4}‎ is stringent, while the Rama is lenient. In the case you describe, even Sephardim (who follow the Shulchan Aruch) have room for leniency due to two additional factors: 1. Some opinions say that bishul akum done in a Jew’s house is permitted. 2 Some say that bishul akum done by a Jew’s hired workers is permitted.‎{&&5}‎
Entering a shul: While it is difficult to find explicit sources on the matter, we see no reason to forbid a non-Jew to enter a shul. The Rambam states that non-Jews were permitted even on a large portion of the Temple Mount.‎{&&6}‎ It is also clearly customary to allow non-Jews into the sanctuary of a shul, all the more so when the need is as clear cut as in this one.