Learn Romanian: Vreau să înțeleg perfect problema. - I want to understand the problem perfectly.

In this sentence, perfect is an adverb modifying the verb să înțeleg (to understand perfectly), not an adjective describing problema. In Romanian, adverbs of manner usually look exactly like the masculine singular form of the adjective, and they never change their endings to match nouns.

Could I change the word order and say Vreau să înțeleg problema perfect?

Yes, you absolutely can. Placing the adverb immediately after the verb (să înțeleg perfect) sounds very natural and puts slightly more emphasis on how you want to understand it. Putting it at the end is also correct.

Why do we use să înțeleg here instead of the infinitive form of the verb?

When linking two verbs where the first expresses a desire, intention, or necessity (like vreau, meaning I want), Romanian uses the subjunctive mood (the să form) rather than the infinitive. Literally, you are saying I want that I understand.

I always forget if problemă is feminine or neuter. How can we tell from this sentence?

problemă is feminine. You can tell here because to say the problem, the final -ă changes to -a (problema). If it were a neuter or masculine noun ending in a consonant, the definite article would usually be -ul or -le.