I agree with both Gray and Jennifer that the meaning is identical, however, in my experience, "are you anticipating any problems at the party tonight?" sounds far more natural to me as I rarely hear questions begin with "do you" over "are you" in my region/social circle. I hope this helped, good luck!

I agree with Gray. I would normally say "do you anticipate," in most circumstances, though, because "are you anticipating" sounds a little more formal.

Both are equally correct, and the meaning is pretty much identical. There is perhaps a very slight difference due to the tense change, but it would be hard to explain, and is not important. You can choose either one.