Yes, with the proper anchors. I would suggest you drill a small starter hole first to ensure you don't crack the tile.
This might work. Take the paper holder to Home Depot and have a HD employee try this. https://www.homedepot.com/p/Everbilt-6-32-tpi-x-1-4-in-Stainless-Steel-Socket-Set-Screw-2-Piece-811938/204274232
There is a plate that you mount to the wall and then there is a small set screw on the bottom of the base that holds it to the mount. Very easy to install.
It swivels in and out, but not around (like a clock). Hope that helps.
Yes, if you fold the towel in thirds. Good idea!
Yes, no problem. The angled design holds it very well, better than some others I have seen.
Please contact Home Depot directly for product availability, pricing, and shipping.
You'll have to ask your local Home Depot