1. This is my cousin David.

2. Please meet my cousin David.
3. Let me introduce my cousin David.

(Do they have the same meaing? Are they the expressions we can use when we introduce somebody to others? Why do we have to use 'meet' in this situation? Can't we use other expressions?)

Yes, the three sentences mean about the same thing. If you don't want to use "meet" in the introduction, your first sentence will be fine.