If a pipe is connected to the damaged blowout preventer in the Gulf of Mexico and can pump in drilling mud then why can't the same connection be used to suck out the oil instead?

You are correct- they can and they are doing it!

The choke and kill lines running off the side of the BOP can be used to produce the oil. BP is actively implementing the "Q4000 Direct Connect" option. Haven't a clue as to how much would flow from these lines.

The oil will go into the Helix Q4000 vessel, and will have to be offloaded. See details from BP in their video presentation. They said on May 31 that it will take about 2 weeks to implement.

I am hoping for a technical briefing from BP on this option and the LMRP soon. Keep checking the BP website.

See related link.