Reading here and there articles on the Internet about good practices when bidding on Elance, I heard this several times about people mentioning their professional URL in their bids. Is it something recommended? Is it actually allowed by Elance?
Yes, why not. Elance will allow you do that as long as your work via Elance. It is recommended!
However, if you think you can outsmart them by finding clients on Elance and then doing the actual work off-Elance, then they will quickly track you down and lock your account.
So use Elance to work and if you don't like paying fee, just attach it to your desired price.
Yes, it is allowed. Actually, showing your work is the only major thing for a client to accept your proposal. You need to have relevant portfolio to show your skills and expertise to a client.