Learn Dutch: Zit gelukkig mee. - Luckily things are going well.

This is an example of conversational topic drop. In spoken Dutch, an unstressed subject or object at the very beginning of a sentence is often left out. The full sentence is Het zit gelukkig mee. By dropping Het, we are left with a verb-first declarative sentence.

What is the infinitive of the verb here, and how is it used?

The verb is the separable verb meezitten. Literally, it means to sit with or to sit along, but it is used idiomatically to mean that things are going your way, going smoothly, or that you are having luck.

If I want to emphasize gelukkig by putting it at the beginning, can I still drop the subject?

No. You can only drop a pronoun when it occupies the very first position in the sentence. If you move the adverb to the front, you trigger inversion and must include the subject, becoming Gelukkig zit het mee. You cannot drop the het because it is no longer at the front of the sentence.

Does gelukkig always mean luckily?

Not always. As an adjective, gelukkig means happy, like in een gelukkig kind. However, when used as an adverb to comment on a whole sentence, like it is here, it translates to luckily or fortunately.