If I delete a file, why doesn't it show up in C:/$Recycle.Bin ?

I have noticed on both of my computers when I delete something, it doesn't show up in $Recycle.Bin even though I had hidden items viewable. Is this normal?

Also, on my desktop, I have noticed that even if my recycle bin is empty, my MBAM threat scans started with C:\$Recycle.Bin. I also notice that various files from subfolders of the system folders are in this folder. Is this normal? (MBAM threat scans don't find anything wrong)

What you are seeing is normal.
For security reasons the files in the Recycle.Bin folder are given random names. There is a separate subfolder for each user account, including built in accounts. This is to prevent looking at the files deleted by another user. They will be restored to the original name.

There should be no reason for regular users to look in the Recycle.Bin folder. It is hidden for a reason. Many of the details of how the recycle bin works are officially undocumented and some of the details have changed over the years in different versions of Windows. Improper modification could corrupt it's contents and prevent recovery of files.

BMAM scans the folder because it doesn't know it is empty until it has been fully examined.

You're not mean to look in $RECYCLE.BIN, you're meant to use 'Recycle Bin'. The first is the folder that hold the index of deleted files for each user, the second is the your personal set of deleted files.

Add or Remove Recycle Bin from Navigation Pane in Windows 10