Learn Romanian: Noi înțelegem problema lor. - We understand their problem.

No, lor means "their" and it is completely invariable. Whether it is one problem (problema lor) or many problems (problemele lor), the word lor stays exactly the same.

Why is the word order problema lor instead of lor problema?

In Romanian, possessive words like lor usually come right after the noun they belong to. English says "their problem", but Romanian structures it as "the problem theirs".

Why is it problema instead of the dictionary word problemă?

Because possessives like lor require the noun before them to have a definite article attached to the end (meaning "the"). For the feminine noun problemă, the final -ă changes to -a to become problema.

Is the word noi strictly necessary in this sentence?

Not at all! The ending of the verb înțelegem already tells us that "we" are doing the action. Native speakers often drop the pronoun and just say Înțelegem problema lor.