The newest versions of the battery are slightly larger in Capacity (5500-5800 mAh) but that won't buy you much for run time. There are quite a few out their but you do need to be careful as some are cheap knock offs which could cause problems! Here's IFIXIT's part: MacBook Pro 13" Unibody (Mid 2009 to Mid 2012) Battery Apple P/N: 661-5557
Sorry you can't put some other heat sink into your system other than the one that was designed for. You may want to do some cleaning of dust build up within the fans and the heat sink fins to improve the air flow. The other thing here is if you are up to it, is to clean off the old thermal paste and apply a fresh coat of a good quality paste. Here's the IFIXIT guide: MacBook Pro 13" Unibody Late 2011 Heat Sink Replacement which walks you though the process of taking the system apart and a list of the materials you'll need.
Sadly, no, you can't swap out your optical drive to put in a dedicated GPU chip. The interface the optical drive uses is not able to support any GPU options here (SATA). The external connection you could connect an external case which houses a PCI GPU card is Thunderbolt2. The best thing would be to sell your system and get a model that has what you want for a GPU.
You face quite a few problems here trying to fit a GPU into your system. You'll need to design your own logic board with all of the needed pieces that would fit in the same space as the optical drive which would include the following:
- Design a heat transfer/exhaust system for the GPU as it does get hot and not interfere with the rest of the system
- Figure out how to power everything from within the system, you'd loose run time which was one of your bug-a-boo's in wanting a bigger battery
- Design a PCI to Thunderbolt bridge so you could interconnect it to the external Thunderbolt port.
I would estimate it would cost you around 50k US between all of the engineering, parts and manufacturing. You also will need software support if you don't use a GPU Apple has already made the needed drivers.
Are you sure its worth doing all of this??
As for the battery: Nothing larger than 5800 mAh that would fit or have the correct connections to connect to the logic board.
Remember Apple never offered a Unibody 13" system with dual graphics. I think you would be happier making the big jump to the new 2016 15" Touch Bar MacBook Pro if graphics & battery life was so important. The newer systems use less power so the battery last longer. In addition you can get external USB-C batteries like Mophie powerstation USB-C to add runtime. With enough Mophie batteries you could be in the middle of the jungle for a week or more!