The Hittites annexing Egypt would be very difficult. Defeating Egypt in a masterful campaign... sure. Even conquering it... I can see that. But holding on to it, that's the trouble. It will simply break away at the earliest opportunity. Considering the logistics of the time, the core of Egypt is just too far outside the direct sphere of influence of the Hittites. An alternative is to basically become a foreign dynasty/elite, while not holding the original Hittite heartland. Best case scenario: a highly succesful invasion/conquest is followed by a Hittite overlord setting up shop in Egypt, essentially becoming the new pharaoh... while also being more-or-less independent from the Hittite state. A relative of the Hittite ruler, in charge of an Egypt that is (at the outset) a Hittite vassal... and later might become more independent.
Although the two timeframes are vastly different, one might compare this to the later Ptolemaic dynasty. This goes very much into your first question: the Ptolemids found out very quickly that they'd either have to adopt a lot of Egyptian cultural practices, or they'd never be accepted. So they just 'went with it', and in the event, that included adopting the Egyptian practice of incestuous marriages in the royal family. The Hittite usurpers, however, will probably adopt certain other practices instead, sincxe they are likely to vehemently object to this one. Hittite laws on sexuality explicitly forbade incest, and demanded it be punished by death. (The Hittites are truly fascinating in their views on what was ethical: they had amazingly strict laws on various kinds of sexual behaviour, often punishing transgressions with death, while certain types of murder were merely fined rather mildly... or even permitted. For anyone studying ethics, the Hittites never cease to amaze.)
Regarding the second question: that depends entirely on the relation between the Hittite state and "vassal state" Egypt. If the Hittite rulers og Egypt want to be more independent, that will cause friction, and perhaps war between them. But it is also possible that they remain staunch allies. It's very hard to predict.