Is sending a shared spreadsheet with the university names, deadlines, and a box to check when completed be something beneficial or appreciated by my recommenders? I thought maybe it would be nice for them to have all the info there and also get to see if they forgot about a particular application.
This is exactly what I did when I applied to grad school. One of my recommenders asked me to create such a sheet, and I shared similar lists with the other recommenders, after checking if they would prefer to have them.
It is a good idea, but ask first, particularly if you want them to check a box after submitting each letter. It might otherwise look a bit presumptuous and as if you were telling them how to organize their work.
It's helpful to get a concise list of the schools a student is applying to and the dates, but putting it into the email request is enough for me. TBH, if the list of schools you're applying to is so long it takes a spreadsheet to keep track of all them, the spreadsheet isn't really a solution to the actual problem.
I'm always delighted to write the first couple of letters for a student because I'm always so happy they're going on to graduate school. I think that's a great choice for most students who did well as undergrads. Once I've written the first LOR for a former student, it's pretty quick to revise it for the next school. But once I've done 3 or 4 letters for the same student, the thrill definitely begins to wear off.
Possibly more helpful than the spreadsheet are the basics: copies of your transcript, CV or resume, any essays you expect to submit with your applications, and suggestions for anything you hope they might emphasize in their letter. Completely optional, a photo can also be helpful for reminding them who you are. (It's also helpful to plan ahead and cultivate future references by participating in class and hanging out in their office hours so they get to know you well enough they don't need to be reminded who you are.)
Also, here in the US, if your reference knows you as a student, pretty much everything they know about you is confidential under FERPA unless you explicitly grant them permission to discuss your academic career and performance.