If you are going so far as to say “would you please”, you are probably better off with the more formal construction.

Then if you add …”two things” it doesn’t really make sense and it could be heard as “do me WITH two things” and then you’re really in trouble (again, unless you like that sort of thing… er, things)

HOWEVER very few people speak perfect English, so you can probably get away with saying “Would you please do me two things” and people will know what you mean. But you won’t impress your English Professor.

Though you might try doing HIM with two things. You never know, he might like that sort of thing.

You could say “Would you please do me two favours?” or “Would you please do me two services?” but these are rather special cases. Doing someone a favour, or doing someone a service are acceptable. Doing a thing is not.