BlockChain scalability: How does BlockChain process multiple parallel transactions?
You should probably read up a bit more, but here's some help:


  miners in the peer to peer network will (individually?) start solving for the next valid hash in the block chain. ( does the peer network use the combined processing power to solve together, or do nodes solve individually independent of each other?)


Nowadays miners work together in mining pools.  These are not part of the actual protocol, but is just the way the mining ecosystem has evolved.  As far as the Bitcoin protocol is concerned, you can look at an entire mining pool as an individual miner.  Each pool is responsible for trying to make their nodes work in parallel.


  Regardless of this, the peer network will be solving for the next hash in the blockchain which will verify 1 transaction


This is where you are wrong.  The hash is for a block of transactions, not an individual transaction.  For many reasons, it is advantageous to do the proof of work on blocks rather than transactions.  See this question for more explanation on that.

I hope this helps.