Why is Bitcoin's unconfirmed transaction count rising and what effect does it have on the network?
Yes, you are correct.
In mempool = Unconfirmed.
Blockchain = Confirmed.

A transaction with 1 input and 1 output is approximately 192 Bytes.
Using 20 Sat/Byte will result in 0.0000384 BTC in transaction fees. However in real-life, most transactions have more than 1 input and 1 output. So on average the fees are much higher. 

If you take a look at Johoe's Mempool Statistics, you can tell that about 1/4 of the transactions in the mempool are sent with 50Sat/Byte or lower. These transactions will probably be stucked for a long time, or dropped gradually when they reach their mempool expiry time(default 2 weeks for Bitcoin Core v0.14.0 and later).


  Will this ever out-scale the number of miners on the network?


Not in the sense that the network will go down, the target solve-time of a block is 10 minutes. It doesn't matter if there are 10k miners or 10 million miners, the mining difficulty will over-time be adjusted so that the average block time approximate to 10 minutes given the total hash rate of the network.

At this rate, where there are more incoming transactions than what the network can process. The network will be more and more congested and eventually a lot of nodes mempool's will be saturated(if they have set a max cap), and new transactions will be rejected.