See instructions, looks like your facet may be addressed in section B https://www.moen.com/shared/docs/instruction-sheets/ins745c.pdf
Depends on faucet. If you have a faucet like section D in the attached instructions from Moen https://www.moen.com/shared/docs/instruction-sheets/ins745c.pdf
It is 7/8 inch diameter by 4 inch long
I used the tool on two cartridges, absolutely no issues. Channel locks provided sufficient grip and the tool didn’t deform at all. Given the price tag of this thing, I have to believe you’d spend a lot more in your “labor cost” to fabricate your own.
Stick with what the manufactor recommends.
You can usually find correct replacement parts locally or on line unless your fixtures were made by a company that is now out of business. If you live in a tract home, disassemble as far as you can and take photos. If it is a faucet with a separate shutoff try to get it apart and take the old parts out. Go to the nearest hardware store with the part (if you have it), show the photos and explain what you want to the plumbing section salesperson. They can usually tell you what you need. If it is a custom home they might not be able to tell or the part may need to be ordered. In this case I suggest shutting off the water, take a stem out and take it to the hardware store, plumbing supplier, etc. to show them what you have and make sure you get the right part. This is an age old problem with plumbing fixtures. Many cartridges, stems and other parts look very similar. Without having the old part out of the faucet and side by side with the new one it is difficult to tell if they are the same.
This is the tool that I successfully used for my roman tub faucets. FYI, the descriptions that HD gives for a lot of the stuff they sell are seriously misleading. I suspect they are generated by someone who has no idea what they are describing. Try to get the manufacturers part number if you can find it, then go to the manufacturer's web site to get a more accurate description. Or, make sure it can be returned and just buy it and see. HD does have a reasonable return policy and I have even talked them into taking back stuff that was a special order because their description of what I ordered was incorrect (but that isn't guaranteed).
I would say no, unless you can design your own invention.
No, It not necessary for installation of showerhead.
how to use it