OK I think I have it figured out. With those command lines, the server has basically no lag anymore. Celebration!
But every so often, when I connect, I have terrible lag. Chat takes 30 seconds to appear, land stops loading for minutes at a time, etc. Again, by monitoring on Putty, I can see that when I enter chat it goes to the server almost immediately, but it takes forever for data to be sent back to me as a client.
I can then disconnect, reconnect, and the lag is gone again. It seems like sometimes when I connect, I just do not establish a good connection, and I don't receive info from the server for a long time. I don't know if this is something to do with my internet connection, it might, but is it possible that Edu sometimes does not establish a good connection with a client and doesn't prioritize to send them data? I haven't really heard of this problem before with vanilla Minecraft, so I'm wondering if it's a weird Edu problem.
It seems to happen about 1/5 times I connect. It has always been fixed by disconnecting and reconnecting (knock on wood). I haven't had students complain about this, so again, there are chances it is just something weird with my own internet. But why are random connections stuck as laggy and bad for receiving data, but send data just fine, and can be fixed by reconnecting? Strange stuff.