Hi and welcome to Switzerland, Baden and ABB!
1) The best way to find a flat in Baden is to use web search engines such as
www.homegate.ch or
www.nabhome.ch
2) There are many outlying small towns and villages with good train or bus connections to Baden. What's nice or not depends on your taste. There is nothing that I would consider REALLY bad though.
www.sbb.ch will give you train and bus times between wherever and Baden.
3) Many places to buy furniture too. There is an IKEA not far away. If you want something more upscale there is the central store for Möbel Pfister about1 hour away. There are quite a few local furniture stores too.
4) The cost of a monthly pass will depend on where you live.
http://www.a-welle.ch will give you details about a local train/bus pass.
Anything else you need to know just holler. What part of ABB will you be joining?