On English grammar: Is it possible to use the adverb "recently" in present tense?

Recently indicates something from the past and using it to refer to something in present tense wouldn't make sense. You sentence should read, "I recently had a problem."

Yes, it is possible, but not in your example. You could rephrase it to say, "I have recently had this problem," or, "I have recently been having this problem." The event that 'recently' describes still must have taken place in the past, but in these sentences, the verb 'have' is in the present tense because it indicates that the past event still holds some significance in the present. If we said, "I had recently been having this problem," it could describe the same event but it implies that it is no longer significant in the present.

The word recently in the past tense. Example: The storm happened recently.

Meaning the storm happened not too long ago but in the past

Using "recently" in the past is grammatically correct.

Recently, a problem developed.

Recently, a problem arose in my life.

Recently, I had a problem.

Recently simply adds the information that the action occurred meaning- "not long ago."