Could someone explain to me why "אבא" is pronounced "ába" but "אמא" is "íma"?
Because they have different nikkud (vowel symbols): אַבָּא and אִמָּא. Of course you can't see that without the nikkud, and Hebrew is almost always written without it, so you have to learn the words and their pronunciation. When writing without nikkud, in many cases the i vowel is represented by the letter י (Yod), but there are exceptions and אמא is usually one of them (although it can also be seen as אימא sometimes).
What confused me also is that a musician I like, Shimshai, has a song called "Abba Amma" so I presumed 'mother' was pronounced thusly.
Abba Amma means "Father Mother" in Tamil (and probably other Indian languages but I'm not sure), so that's probably where that comes from.