Why is Bitcoin's unconfirmed transaction count rising and what effect does it have on the network? I see that blockchain.info is reporting a large and steadily growing number of unconfirmed transactions.

I assume that these are all transactions broadcasted to the network but never confirmed as a result of low fees.

What does this mean long-term? Given my understanding of the blockchain, only transactions on the blockchain are considered confirmed and don't exist on the blockchain if they are not confirmed - and instead, exist only in the mempool.

According to blockchain.info's mempool size chart, the mempool size has increased considerably since genesis, and some initial research shows that the size does grow with abnormal network conditions (high volume spikes, etc).

At the time of writing, the number of transactions is 285,565, totaling about 228.71450055 BTC that are tied up in fees.

That means about 0.0008009192322 BTC per transaction on average. However, following the rule of 20 satoshis per byte, on average, each transaction would need only 0.00002 BTC in fees - about 40 times less than what is currently tied up in BTC fees.

What's going on here? Will the number of unconfirmed transactions die down over time? Will it increase uncontrollably? Will this ever out-scale the number of miners on the network?

Or am I completely missing something here?
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.