This is where I get lost as well. I think I may have an inkling of whats going on now, and please correct me if this is wrong.
Will do.
"Mining" is simply the act of using your computer to solve the math of blockchains.
Yes, but I dislike the word "sovle," becuase it's really just doing the same thing over and over and trying to get lucky.
Blockchains are what Bitcoin uses to verify and secure transactions.
Very close. Blockchains are where the ledger of secured transactions are kept.
If your computer solves a blockchain math problem (bitcoin discovery), you're rewarded with a brand new bitcoin, or some fraction of a brand new bitcoin. Not sure which.
Close, again. If your equipment solves it and you're mining solo, you get 12.5 bitcoins. Most people mine as part of a pool and share in the proceeds when anyone in the pool hits, so they get fractions. Odds of hitting when mining solo? Damn close to zero. That's why people mine in pools.
Also, your computer can't mine for Bitcoin. It's only mineable (profitably) on specialty equipment - ASIC miners - which are expensive and hard to get. You can mine Bitcoin Gold (a variant designed to exclude ASIC miners and run on computer GPUs) or some other cryptos on your computer, though.
Mining bitcoin is "free," beyond the cost of equipment and electricity to run it.
Yes.
Bitcoin that have been "discovered" are what people are using for transactions, or buying and selling on the open market, causing the fluctuations in price.
Yes. Every Bitcoin in circulation was originally mined. When the second block was mined in 2009, the first 50 bitcoins were created as the reward. (The first block, the Genesis Block, intentionally burned the 50 bitcoins that would have been the reward.)
Price fluctuations are entirely market-driven.
There are only 21 million bitcoins in existence, and the discovery of new ones gets harder as more are mined. This is by design.
No - there will be almost 21 million max, but they haven't all been mined yet. About 16.83 million have been mined so far. The per-block reward is now 12.5 bitcoins; it gets cut in half roughly every four years. Last coins will be mined approx 2040, but the mining reward will be trivially small long before that.
So new coins appear less frequently over time... but they are not "harder" or even "discovered" in the sense that the reward happens every ten minutes, on average.
The difficulty and processing power required to solve blockchains these day is beyond the capabilities of a personal computer, so it's mostly being done by large server companies.
Not quite. The existence of ASIC miners has driven up the difficulty for Bitcoin mining, and ASICs are only made, sold, and distributed by a tiny handful of players. You either have an in or pay exorbitant amounts. If it weren't for that, it would still be accessible to individuals. But ASICs don't work for everything. They work on the specific algorithm used for Bitcoin's Proof of Work. Different Proof of Work algorithms can't be done on ASICs. When Bitcoin was created in 2009, there were no ASICs. That's actually the impetus behind the formation of Bitcoin Gold last year, which uses an algorithm that can't be mined on ASICs.
So my question is:
Since the security and verification of bitcoin transactions are reliant on people mining bitcoin, what happens to the currency when either all bitcoins have been mined, or the difficulty is no longer worth the effort? How do the blockchain problems get solved when not enough people, or nobody is mining bitcoins? Does the currency just become worthless?
No, the process also includes a mining fee which people can voluntarily add to their transaction to "prioritize" it. Whoever mines the block gets the mining reward PLUS the fees attached to all the transactions mined.
When the block is crowded, this is currently used to get priority (or else you may wait many blocks before your transactions is confirmed in a block). When the mining reward eventually gets low, the addition of small fees to transactions will substitute for the reward.