but I don't exactly see it that way because each mining node is competing against all the others.
There is no competition. Every hash found has an equal chance of matching or exceeding current difficulty, which is around 39.5 Billion.
If there's no competition, then why am I not solving ALL of the blocks?
Besides this, more importantly, each hash (ie difficulty of 1) has the same probability of solving the block as 1/1024 of a miner hashing at difficulty of 1024 (accepted share with difficulty 1024)?
And when my miner has difficulty set, there are no accepted shares below that difficulty. So I'm assuming that 934/1024 doesn't show up, because it's not accepted. My understanding is that 1024 is a larger fraction of 39.5B to be solved.
Competition is not against the other miners, but against... Well... Probability. There's no competition actually, only probability. A hash of difficulty 1 has no chance of solving a block of difficulty 39B. A hash of diff 39B or more can "solve" it. Your miner submits hashes to a pool just to prove to the pool that your miner is working. Otherwise, you shouldn't have to submit shares below the current network difficulty. When you solo mine (not in a pool), you don't need to submit hashes below the difficulty. Pool diff and network diff are two different things about the same concept of the "target".