Learn Dutch: Wij ontmoeten hem morgen. - We are meeting him tomorrow.

Just like in English, pronouns change depending on their job in the sentence. Hij means "he" (the one doing the action), but here he is the one receiving the action. So we use the object pronoun hem, which means "him".

I just learned that mij can be shortened to me. Does hem have a short version too?

Not in writing! While mij and jou have official short written forms (me and je), hem is always written as hem. However, in fast spoken Dutch, you will often hear people drop the "h" and pronounce it like "'m".

Could I change the word order and say "Wij ontmoeten morgen hem"?

It's much better to say Wij ontmoeten hem morgen. In Dutch, when the object is a specific pronoun like hem, haar, or mij, it almost always comes before time words like morgen (tomorrow).

The English translation says "are meeting". Why isn't there a separate verb for "are"?