RAM upgrade for MacBook (13-inch, Aluminum, Late 2008)

1. Your MacBook can actually address as much as 8 GB. It must be PC3-8500 (1066 MHz). Nothing else will be reliable. Do not purchase memory from any source other than a vendor known to support Macs. Crucial and OWC / MacSales are the only two that I can recommend. To determine if your Mac's performance is being constrained by available memory use Activity Monitor: Use Activity Monitor on your Mac - Apple Support. If "memory pressure" is often "red" then additional memory will be beneficial.
2. Probably. Microsoft and Google products are notoriously inefficient.

I do not believe that Mac's battery is available from Apple any longer. You will need to find an aftermarket vendor. Consult OWC for a replacement.

John Galt said pretty much everything I would have recommended. You can get Crucial brand RAM from other vendors so it may be worth shopping around.

If you go for 8GB RAM you must do a firmware upgrade first. This may have already been done as part of your earlier upgrade to Yosemite but oddly enough I had Mavericks on a computer I was given and it had somehow avoided the firmware upgrade. The OWC/mac-sales.com web site has information about this.

On the computer I was given it had Mavericks but original 2 GB RAM. Activity Monitor showed Mavericks was taking 1.98 of those 2 GB just to run the OS, leaving 0.02 GB RAM for everything else! When this happens your computer starts using the 5400 rpm hard drive for RAM which is slow, and the original hard drives are very slow compared to 7200 rpm models which this makes it doubly worse. You can install a faster and larger hard drive but at some cost of higher power demand, or if you don't mind not having a big hard drive you can go for a real speed boost and install a SSD. You can even have a SSD and a standard hard drive if you don't mind using the space taken by the optical drive. One nice thing about the late 2--8 MacBook is it is pretty expandable.