No, it doesn't. 'A big mouse' refers to some mouse which happens to be big. 'The big mouse' refers to one specific mouse.
For example, if you were talking about your pet mice you would say 'the big mouse' because you have a specific mouse in mind (i.e. your pet mouse). I hope this makes sense.
Not a real exception, just a rule you did not know.
Feminine nouns do not have to end with -a, there are many feminine nouns that end with a "soft" or "semi-soft" consonant.
I wrote a bit about genders in Polish https://www.duolingo.com/comment/14133935 and my opinion on those is :
If a word ends with ń ć, ś, ź, ż, sz, cz - you can toss a coin, or check a dictionary, it might be feminine or masculine.
There are Tips and Notes until some point, not for the whole course. More are coming. You can access them by clicking at a skill icon and then the lightbulb icon, if you don't see it in your app, use the website version.
Also, you may find a lot of interesting topics here: https://forum.duolingo.com/comment/16296174