What is the reason for the lack of response to prayers for winning the Green Card Lottery (Diversity Visa Lottery) to America, and is traveling to America incompatible with prayer, or does prayer not change destiny?

It is not permissible for a Muslim to emigrate to and reside in lands of disbelief unless there is a necessity or need, due to the potential for evil, temptation, and corruption therein. The non-response to a supplication for emigration may be a mercy from Allah, because one's religion is a Muslim's most valuable asset, which must be preserved, and many Muslims may deviate due to residing in those lands. The non-response may also be due to the fact that what Allah chooses is best, as He Almighty said: "But perhaps you hate a thing and it is good for you; and perhaps you love a thing and it is bad for you. And Allah knows, while you know not."