It's more natural to say "getting" than "going." If you say "going," it sounds like you're describing the way he has been going... for some reason. Even then, it doesn't really make too much sense.
Nope, neither are correct.
Are there any problems? / Is there a problem? (You can't use "any" with a single "problem")
"There is any problem" is just wrong. You can't ask questions like this. It's exclusive to Russian (between English and Russian at least).
You could turn it into a statement by removing the question mark and fixing the sentence like so:
There is a problem. / There are problems. (You can't use "any" here.)