Is Buddhism compatible with theistic religions such as Christianity and Judaism?
It depends on whose Buddhism, and whose Christianity or Judaism. The ultimate concern in Buddhism is “suffering” and the cessation of suffering. There are many today who find Buddhist approaches enriching. Since the focus in Buddhism is practice and experience rather than belief or doctrine, it has the capacity to adapt to many different cultures, belief systems, and circumstances.
Does Buddhism help alleviate stress, anxiety, loneliness, and other problems of modern life?
It may and often does, but that is not the aim of practice. To practice the Eightfold Path is to cultivate an understanding which transcends the usual Western concepts of self-improvement. Buddhist practice aims at awakening to the true nature of experience, which is emptiness, rather than striving to attain or avoid any particular experience.
Is belief in reincarnation a requirement for practicing Buddhism?
No. Buddhism is not oriented toward belief or doctrine. The only requirement for practicing the Buddha-dharma is a willingness to apply and evaluate the recommended practices in your own life.
There is some disagreement in Western Buddhism about the topic of reincarnation or rebirth, and a number of interpretations of this teaching may be found. There is the literal traditional view that the karma one generates in this life will determine the realm into which one will be reborn, as a god, human, animal, hungry ghost, or hell-being. A psychological view would understand that we are reborn in each instant and that our present actions determine our psychological state in future moments. Finally, a philosophical view would see rebirth as simply the total conditionality of all existence. One can fully practice the Eightfold Path while holding any of these views or none of them.