Learn Dutch: Moet veel werk inhalen. - Have to catch up on a lot of work.

The subject Ik (I) has been dropped. In casual spoken Dutch, an unstressed subject at the very beginning of a sentence is often left out. This creates a verb-first declarative sentence, which sounds very natural in informal contexts like text messages or quick chats.

Could the dropped pronoun be something other than Ik?

Yes. While "I have to catch up on work" is the most likely meaning here, conversational topic drop simply relies on context. If you were just complaining about a colleague's backlog, this exact sentence could mean "He has to catch up on a lot of work." The listener knows who you mean based on the previous conversation.

Inhalen is a separable verb, so why is it written as one word here?

Because the sentence uses the modal verb moet (must/have to). When an auxiliary or modal verb pushes a separable verb to the very end of the clause, the separable verb stays intact as a full infinitive, rather than splitting into parts.

If I want to emphasize veel werk by putting it first, how does the sentence change?

Moving veel werk to the first position triggers inversion (putting the verb before the subject), so it becomes: Veel werk moet ik inhalen. If the "lot of work" was already the topic of conversation, you could apply the topic drop rule to the object instead and just say: Moet ik inhalen.