Confused with Bidding in Contract Bridge
The bidding process continues until all players have passed in succession, or three players have passed in succession after a bid.  You seem to misunderstand the double and redouble bids - they can't go on indefinitely.  After a double from the opposing team, your only options are to go to a higher level or redouble.  After a redouble from the opposing team, your only option is to go to a higher level.  (Or pass out of course, in either case.)  If bidding reaches 7 no trumps redoubled, there is nowhere left to go.
It would be easy to assume that "double" and "redouble" imply x2 or x4 value for the contract - much too easy!  In fact doubling and redoubling also do some rather complex things to scoring overtricks or undertricks in the final score, dependent on the "vulnerability" of the contract, that has little to do with a simple multiplier.  And that's before we get into the additional "insult bonus" for successfully making a doubled or redoubled contract... Here's a link to the Wikipedia sections on the subject, probably better and conciser than me trying to outline all of it from memory.
Declarer is, as you say, the first player to mention the trump suit (or bid no trumps) in the partnership that ultimately wins the contract.
Unfortunately dummy has little to do except twiddle his thumbs until the contract is played out.  Fortunately Bridge is a complex and tense game that is usually as much fun to watch as play, and generally I don't find it a big problem to sit there watching partner play the hand, even online.  You definitely want to be able to berate your partner after she's finished making a hash of the hand, with useful advise about how you would have done it!  Er, I mean "commiserate with", not "berate", obviously...