I decided to finally start learning z80, so I will learn like this.
1) Mimas for on-calc.
2) 28 Days
Apologies for sparse topic, will expand tommorrow when I wake up.

Do you not have computer access to use something like Brass + the Doors CS SDK for this? Anyway, we look forward to your questions, and hopefully you can give some feedback that will help inform the TI-83 Plus ASM in 28 Days improvement project.

I simply like the ability to program on-calc. That is why Axe always appealed to me since I could program on-calc.

That's fair enough, and it's the reason I like to promote TI-BASIC as a beginner programming language. You have everything you need in your pocket whenever you want to do a bit of programming. Anyway, let us know how things are going.

This is more of a general z80 question, but still: What does Garbage Collecting do? Is it like a way to free up data that was improperly deleted?

No, it is more akin to how the Recycle Bin works on Windows computers. The data is simply marked as deleted, but not truly removed. garbage Collecting compacts all the existing, non-deleted items in your Archive, leaving all the free space at the end. Until that happens, there are a bunch of holes with deleted variables that can't yet be used for new variables. It is very similar to defragmenting storage.

Since i'm currently on my way to code a defrag routine, i did some tests to see what the TI-OS really does when garbage collecting.

First, it is interesting to note that it actually defragments applications.
This is not supposed to happen, cause it is normally done automatically when deleting an app.
But when an app is deleted using the _deleteapp rom call, it is just marked as deleted, and the applications area is not defragmented.
In that case, runing a garbage collect will do that job.

Secondly, the variables area is defragmented on a per-sector basis.
For example, let's say you have 3 contiguous sectors, all holding variables.
If all the variables in the second sector were previously deleted, the garbage collect will erase that sector, but will not always move the existing variables from sector 3 to sector 2.
That is something to keep in mind when reading the archive space, cause as far as i'm concerned, i initially thought that once you find an empty sector, there cannot be any more variables in the following ones.