Your system has two memory slots. For best performance, the DIMMs that you install need to be as close to identical as possible - same size, same speed, etc. While it is possible to use two DIMMs that differ in size or speed, remember that the two DIMMs will run at the slower of these two speeds and operation will be slower because of the size differences as transfer operations cannot be interleaved. This is why you see memory manufacturers selling DIMMs in pairs (i.e., two per package). It is much better to purchase a two-pack as the DIMMs have been tested together and matched to ensure this. If you need 8GB, better to get a package with two 4GB DIMMs than a single 8GB DIMM.
In your case, you have one 8GB DDR3-1600 DIMM and one 2GB DDR3-1333. The system will run much better if you remove the 2GB DIMM. You will have more memory and it will be running significantly faster (8GB at 1600MHz instead of 4GB at 1333MHz). Eventually, when you are able, you could then purchase a second 8GB DDR3-1600 DIMM (hopefully a DIMM from the same manufacturer and with the same specs) and gain the double-benefit of having faster, interleaved transfers and twice as much memory overall. Note, however, that I am not recommending that you do this.
Ok, let's talk about BIOS. It sounds like you plan to stick with the Sandy Bridge (2nd gen Core) processor and do not plan to add support for 3rd gen Core processors. That's good, it simplifies the update process. But, this update process is fairly complicated regardless. You will have to install a total of seven BIOS updates and some of these updates will have to be installed using the more-complicated BIOS Recovery process. Are you prepared to do this?
My personal opinion: If removing the 2GB DIMM solves your short-term problem, I would leave the system as it is. It is so old that it would be better, when you are able, to purchase a new and much more-modern system (i.e. new Processor, new Motherboard, new Memory).
Let me know if you want to take on the BIOS upgrade. I can provide the files and detail the process, but I warn you that this process is fraught with potential problems and may brick the system permanently if something goes wrong.
…S
