What is the correct form:
"Without any problem" or "Without any problems"?
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Either will do. It's actually pretty amazing just how interchangeably they're used:
Any means one or more, which means that both options mentioned by you, are correct.
Normally one would just say without problem, skipping the any altogether. It doesn’t really add anything to speak of, and just makes the phrase longer.
But I certainly wouldn’t call without any problems (or with no problems) ‘wrong’.
Also, you can swap in trouble for problem in all those phrases, and you get the same answer; think also of no trouble found.