Bitcoin has moved away from the Longest Chain Rule to the blockchain with the most cumulative Proof of Work as what the nodes recognize as valid blockchain.
I hear a lot about hash rates. How are aggregate hash rates confirmed/reported? Does a miner submit every single attempted hash to the network as a potential solution or do miners self-report their total hash rates?
It is not necessary for miners to report any stats. Hash rate is simply inferred from the average time taken to find a block, multiplied by the current difficulty.
Every block hash has a target it is trying to be under. For example, if the target is "0000008dab3", then when you are hashing your block, you need to come up with a hash that is below that number. When the network's hash rate goes up, people are producing hashes quicker so a hash is found below that target well below 10 minutes (what the network tries to average around). If hashes are found in less than 10 minutes for too long, then the target gets lowered even further, say to "00000000ab3" to require everyone to produce more hashes so it takes longer.
Now to actually answer your question: if you take 2 bitcoin blockchains that stem from the same origin block, but blockchainA has 20 blocks with a very easy target "fffffffdab3" and blockchainB has 1 block with a very difficult target "00000000003", then blockchainB has actually done more work than blockchainA, even though it has less blocks (it is "shorter"). So blockchainB has the most cumulative proof of work.
Because the "difficulty" rate (essentially the number of leading 0's needed for the proof of work) can change dynamically, the longest chain is not always the chain with the most work.
You could, but you'd be competing with everyone else that's also making transactions so you'd need an incredible amount of computational power to outstrip that.
And if you had that much hashing power, you'd be better off using it for mining. It's explained in more detail in the (surprisingly accessible) bitcoin whitepaper.
