Learn German: Alle verstehen das Problem. - Everyone understands the problem.

In English, everyone is singular, but in German, alle literally means all and is treated as a plural subject. Since it acts like the plural they (sie), the verb gets the regular plural -en ending.

Are there other plural pronouns that would use the same -en ending here?

Yes! The plural pronouns wir (we) and sie (they) also take the -en ending. So you could say Wir verstehen das Problem (We understand the problem) or Sie verstehen das Problem (They understand the problem).

Since das Problem is the direct object (the thing being understood), why doesn't the word das change?

Only masculine nouns change their der word to den when they are the direct object (the accusative case). Because Problem is a neuter noun, its matching word das stays exactly the same whether it is the subject or the object.

What happens to the sentence order if we want to emphasize das Problem by putting it first?

In German, the verb must always stay in the second position. If you put Das Problem in the first position to emphasize it, the verb stays second, and the subject moves after it: Das Problem verstehen alle.