Batteries of that vintage are known to swell and damage the case if not not caught or leak into internals. Keyboard mis-clicks are an indication of this.
Coconut Battery is very good product because it tells you the health of the battery ... when 90% charged is really 5% of original capacity.
Apple Stores no longer officially service the hardware of 2011, but may make an exception. AASP (Apple Authorised Service Providers) do not care about the year of the system, and OtherWorldComputing (macsales.com) sells the proper battery for self-service ... easy to do in my opinion.
ifixit.comhas many DIY guides for operations of all vintages of Macs, from memory swap (usually easy) through screen replacement (if one is so bold!).