I'd say that using "cheers" is exactly the same as saying "thanks". Thus everything else can also be applied to thanks.
It ("ta") would usually be pronounced as tar, and so rhyme with bar and car. But in certain accents (e.g. South Wales?), it may be a short a as in tap. This use and pronunciation of ta should not be confused with ta-ta or ta-ra, which is used very commonly, mainly in the Midlands and North of England and in Wales, for good bye. I use cheers, ta and ta-ra quite a lot!
