why is it that in most all public restrooms and some home restrooms that the toliet paper is always fixed to the wall on the right? what is the reason and origins for this?
Probably because most people are right-handed?
Actually, in my last two places, the TP has been on the left…
Try getting your right hand at the roll mounted on the right side of the stall and doing your business. Not too easy without twisting yourself.
But, they’re nicely located for us lefties! 
my TP is on the left.
Do you mean when you’re facing the toilet, or sitting on it?
When you’re sitting, the dispensers in the bathrooms in my dorm vary. The ones at home are on the left.
And just to answer that other important question, toilet paper should be positioned so it slides down from the back, not from the front like some heretics seem to think.
Yes, if only to prevent the cats from sitting on the toilet seat and unrolling the whole thing with their paws.
The toilet paper dispenser (in a multi-stalled bathroom) is positioned as follows:
The first stall will have it on the “pre-fab” wall…not the brick wall. The next stall will have it on the other side of that “pre-fab” wall. They will alternate thusly down the line. That way, you don’t have to poke holes and screws in every single pre-fab wall.
And the toilet paper should ALWAYS go against the wall. And it’s mostly because of the cats, although it allows for a smoother tear-off.
In my house, it’s on the left. Mounted to the cabinet that the sink sits on.
In my youth, I always installed the roll with the paper hanging toward the wall. Then, after college, I took a job involving extensive travel and I noticed that every hotel/motel in the world has the paper hanging on the outside (or at least the several dozen hotels I went to).
I figured they were the professionals, so I adopted accordingly and now have the paper hang down the front. None of these hotels had cats of course, and neither do I.
Yeah well, we have the toilet paper hanging down the front. Sometimes Mrs Skogcat even folds the end into a neat hotel-style triangle if she’s in a silly mood. No cats here anymore and the dog’s idea of mischief is rubbing his dogfood-lips on the beanbag after he’s been eating.
of course i mean when you are sitting on the toilet
“Already in Use”
at my house the tp is mounted to the right of the toilet against the wall that the toilet is against.
two dogs, no cats(dead) 
This makes no sense at all, unless you like constantly having to fumble underneath the roll for the beginning of the paper. In which case I guess it would make a weird twisted sort of sense.