Learn Dutch: Neem twee druppels per dag.

You could, but it would completely change the meaning! Per dag means "per day" or "every day". If you were to say Neem twee druppels om de dag, it would mean "Take two drops every other day".

In English, we often say "two drops a day". Can I translate that literally into Dutch?

You shouldn't translate the "a" literally here. While English uses "a day", Dutch strongly prefers per dag for rates and frequencies. You might occasionally hear op een dag in casual speech, but for instructions like this, per dag is the standard.

Does druppels refer to both liquid drops and the famous Dutch licorice candy?

No, druppels only refers to drops of liquid (like water, liquid medicine, or eye drops). If you are talking about the black licorice candy, the word is drop (or dropjes for individual pieces).

Why is the verb neem used here instead of neemt or nemen?

This is the imperative, or command form, used because you are being given a direct instruction. In Dutch, the imperative is formed by using the stem of the verb (neem) without a subject pronoun and without adding a -t.