Is it time to change the battery?

I have had my laptop for nearly 2 years now and my battery is starting to act strange.

First, it will never charge to 100% when plugged in but when I remove it, it will say it is 100 for a few minutes before going back to the actual %.

Second, the status indicator on my laptop which is an asus n56vz is always red when plugged in. Is this something I need to worry about.

Thrid, I don't know if this is to do with the battery or a collection of dust, but my cpu seems to get hot when watching youtube videos in 1080p. This happens no matter the browser I use.

If it is time to change the battery, can you let me know the cheapest place to get it from. I am in the UK so if it is possible to get a cheap deal in a store that would be good, but don't mind using sites like amazon.
 

It sounds like it is definitely time to replace the battery although the CPU fan might have gone out as well.
 
Are you sure my laptop has a cpu fan? I thought it has one fan for both the cpu and gpu.