No, a router/hub cannot handle POTS lines (Plain Old Telephone Service). Think of phone lines as electrical lines, as they essentially are. They carry voltage - bad idea to connect them to computer/networking equipment!
cat5e is just fine for carrying POTS. I doubt your telephones have rj45 jacks on them, unless they are for a PBX system. Your ADSL modem should have a RJ11 "out" jack that is meant to plug a normal phone into. I suggest not using the out of the modem, but to take in your service directly off the demarc into your wiring closet, and doing your voice/adsl splitting right there.
From your wiring closet/board, connect the rooms/walljacks you want to the phone line. There are a number of patching methods to use to do this. Most common is likely a "66 block" that looks like:
http://therub.org/howtopunch/images/66_block.jpg
Basically at the top you connect the phone line coming from your demarc (the phone company) then at the bottom the wires that run to the various walljacks in your home.
Be sure to label the jacks in each room for which are data, and which are voice. Standard rj11 jacks generally work in rj45 sockets, so it should "just work" if punched down correctly and you use the right wire pairs. However, you may want to simply create some custom rj45 -> rj11 cables if you have the capability for a better fix.
In short.. what you want is fairly easy once you understand how it works
A bit tough to describe on a message board though.
This is of course assuming standard phone service which I presume is true. Otherwise the questions wouldn't make any sense
-Phil