Would more Full Nodes help scaling and transaction speed? There seems to be lots of disagreement between Core, Classic, etc on how to scale up Bitcoin. 

As far as transactions per second, and transaction fees, would simply adding more Full Nodes to the network help solve several me of these growing pains?

It seems to me having more Nodes verifying transactions would both speed up the network and reduce transaction fees. Although, I don't seem much discussion anywhere about this
No, the number of full nodes is decoupled from capacity and fees.

Transaction capacity is a function of transaction size and blocksize. Blocks occur roughly every 10 minutes and have a fixed size. Transactions also have a more or less fixed size, so capacity is currently not going to change unless either transactions get smaller (e.g. Schnorr signatures, MAST) or blocks get bigger (e.g. SegWit, blocksize increase).

Fees are a function of supply and demand of blockspace. Since the number of transactions that can be confirmed is limited over time, transactions with higher fee per byte get preferred. Especially when the network is unlucky and blocks are slow for a period the transactions waiting to be confirmed may build up intermittenly, which makes the average fees in blocks peak.

Full nodes are still important: Besides allowing their owner to trustlessly use Bitcoin, they serve as information providers to thin clients. If their number would be much smaller, the full nodes would be overwhelmed by these requests.