Learn Portuguese (Portugal): Quantos problemas têm eles?

This is how Portuguese shows that the verb is plural! For the singular 'he/she/it has', you write tem. For the plural 'they have' or 'you all have', you write têm. The accent visually marks the difference between the singular and plural forms.

Problemas ends in '-as', so why do we use the masculine word Quantos instead of Quantas?

Even though it ends in 'a', problema is a masculine word (o problema). Many Portuguese words that end in '-ema' or '-oma' (like o sistema or o idioma) are masculine because they originally came from Greek. Since the noun is masculine plural, we must use the masculine plural Quantos.

Why is the verb têm before the subject eles? Don't we usually say eles têm?

In Portuguese questions, it is very common to swap the order of the verb and the subject to show that you are asking something. However, it is not strictly required. Saying Quantos problemas eles têm? is also perfectly correct and natural.

Can I leave out the word eles and just ask Quantos problemas têm?