Is this sentence grammatically correct?

I can problem-solve. You are using a noun as a modifier, so you need the hyphen. Otherwise, the syntax is incorrect. We do not inject the object of a verb in front of the verb, except in a fairly archaic form or for artistic purposes. If you wish to claim the reversed verb-object order as valid, you still need to respect the noun use rules, and grammar imposes a plural or the use of an article (a or the).

It is ok to say, "I can a problem solve", although it is generally considered awkward.

But back to the question, it is pretty clear that you intend to be using "problem-solving" and its equivalents.

It is grammatically correct.

I is subject, can is the verb, problem solve is the direct object. (In this case can is a verb which means means: I am able to) (problem solve becomes a noun phase describing a skill). However, you won't find problem solve in the dictionary, although it is commonly used.

A better structure is: I can solve problems. (in this form can solve is the transitive verb, problems is the direct object).

It would be better written " I can solve problems"

Yes.

Yes it is.