There is a LOT of stuff that happens in the background that rclone does not show you by default (because as you said - the output becomes so messy that is very tedious and confusing to read).
But yes - all rclone transfer types: move, copy, sync all calculates a hashsum on the read-in of the file while it is sending - and then when the transfer is done it checks it's own calculated hashsum to the hashsum the server provides. If they are not identical then it will discard that attempt and re-try.
It just doesn't bother to show you all this background stuff unless you use the extremely verbose debug-log (-vv). I am not sure it it maybe also shows file-hases for transfers using just -v / --verbose. This is not quite as bad to read as debug-level.
First of all I don't recognize the software you are showing screenshots of...
When you said "rclone browser" I assumed you mean the fairly recently added rclone webGUI - but this looks very different.
Are you using that very outdated third-party rclone browser that stopped being developed a long time ago? You should probably link what you are using so we are on the same page before I attempt to answer your questions about this...
Try just using --verbose or -v. This still gives a ton of information, but it still far easier to read. I am not sure if it shows the hash values spesiifically, but give it a try. Debug-log is really only meant for developers to debug the code. It is far too messy to be practical in regular use.
Correct. But also if your drives are not reliable it is an even worse idea to force --checksum because it would stress them unnecessarily with lots of reads.
Definitely look into using --backup-dir like asdffdsa suggests - because when you have unreliable drives you can never be sure that they don't read your files wrong and you then end up coping this mistake to your backup. If you use --backup-dir then you will keep a copy of any "deleted" files, including those that rclone would overwrite or delete in a copy or sync
Gdrive also has a 30-day trashcan for all deleted files that you could recover from in case of emergency, but backup-dir can be your primary safety mechanism without any automatic emptying - and also in the case of a disaster it is much simpler to restore from than picking through the Gdrive trash which can be very tedious...
After you have read the documentation
https://rclone.org/docs/#backup-dir-dir
You can check out here a script than asdffdsa and me created for another user that shows how --backup dir can be used to create dated versioning of all changes that are made.
Then I suspect that this program communicates independently from yor mount (which would make the most sense). In that case it will need to use it's own flag options yes... but please find out what exactly this software is. Either it may be super outdated (I know there existed an rclone browser before) or it is something new that I am now aware of and might be interested in learning more about 