1) Is it true that the practice of blowing out candles on a birthday cake may stem from avoda zara practices? Is it permitted?
2) How about “making a wish” — making a wish and then blowing out the candle surely seems strange to an orthodox Jew! Is it permitted?
You should not do any of these practices. You can have a birthday cake, but without candles and without blowing them out. You are right this practice has pagan and avodah zara roots. Having candles on a birthday cake is a Christian practice in the church to blow out candles according to the age of the person. There is another source for this,. not much better. In Greek times, people on their birthday would try to find favor in the eyes of the moon goddess, Artemis. In an attempt to do so, they would make round cakes (to symbolize the full moon) and light candles on top (to symbolize the light of the moon.) They would then ask the moon goddess to grant their requests. Therefore blowing out the candles and even lighting them should not be done.
V’yitzbar Yitzchok 1-51, RJJ Journal of Contemporary Halacha 54 pg. 73.