For removal - push as though trying to do a poo/give birth (not as hard of course, but if you've had a vaginal birth pushing out a little cup suddenly seems 100x more doable in comparison)
Use stem to grab initially. Do little pushes to move it down while holding stem so that it doesn't escape back up. Once you can reach the bowl of the cup, grab and pinch that instead so that the seal breaks and the top ends up more of a flat 0 or loose 8 shape, then you can pull it out. I tend to tip mine out at an angle so the edge facing my back comes out first, and position my fingers side to side, so that the part I'm guiding out is the thin end of the 0. It can be uncomfortably stretchy if you're trying to remove the entire rim at once.
If you have an actual Mooncup brand they are some of the bigger and stiffer on the market so you might find removal easier with something smaller or softer. I find my Meluna medium to be much more comfortable than the Mooncup. However we are talking millimetres difference and for most people it's fine.
If Ocado is selling one it should be fine. It is the cheap Chinese ones on amazon and ebay you should watch. Otherwise Precious Stars Pads on youtube has reviews of different types, maybe your DD could look there. Most brands can be bought online with home delivery. If she is a teenager and hasn't finished growing and/or isn't sexually active she might prefer a smaller cup than a standard one so PSP is worth a look.
I have a tilted cervix and it's not an issue as it sits in the vaginal canal, it doesn't nestle the cervix like a diaphragm. Emptying at work is OK, you do sometimes get blood on your hand (only one usually) but I would wipe with a tissue so not dripping. Sometimes it's still visible around nails etc but if it bothers you you can make that hand into a fist to hide nail area until you get to the sink, and it washes off easily with soap and water. I don't think anybody would notice. And yes unless you have very heavy periods, you can often empty in the morning and when you get home and not worry about it at work.