Can other people put your contacts in your eyes?!

Technically , she can.
You just have to be very careful because you wont be familiar with her touch.
Nothing serious will go wrong but she might push too hard, or you could flinch, other than that, its fine :)

If you can't put them on and remove them yourself, you shouldn't be wearing them .

What happens if you get something in your eye and have to quickly take your lens out, and your sister isn't there ?

There are ways to do it with long nails by using the sides of your fingers.

Optician

I don't suggest it. Do you plan on her putting them on your eyes every day? I feel like that puts a risk of an eye injury if she isn't aware of something that you wouldn't do yourself. It's really not a good idea.