"Do/Does" is typically used in questions only, but you can use it in a sentence to highlight or emphasize specific information. In your example: "Paul does enjoy surprises, just not on his birthday!" The fact that Paul enjoys surprises is emphasized in the 1st clause. The 2nd clause introduces a contrasting statement. This kind of structure is common. Here is another example: "I do like sweets, but cake is too sweet for me." There are other contexts in which you can use "do/does" like this. Just remember that when you DO (like here ;D ) see it, that it indicates an emphasis. I hope this helps. Let me know if you need anything else.

In the expression you're using its most likely an affirmative to a question. I.e. "Does Paul enjoy suprises?" You could respond "Paul does enjoy suprises.". Though you are correct in thinking it isn't necessary, you cpuld simply say "Paul enjoys suprises as well."

If you use "do/does" in a statement its purpose is to emphasize what you say: - I do enjoy reading = I really enjoy reading