you can buy a ticket for anyone.
you can use your award miles to get a ticket for anyone.
if you buy a ticket for someone, they earn miles for flying, but if you used a mileage-earning credit card to pay for it, you can earn miles for the cost of that ticket.

For example, it is $500 to send her from A-B. You use your Citibank Mastercard that earns you miles on American Airlines to buy her a ticket to fly on Delta.

You earn 500 miles for making a $500 purchase.
She earns miles for traveling on Delta on her Delta account.
The two matter not together, for they are separate things.

If you have many Delta miles and you wish to use them to get a FF award for travel, you can tell the airlines to book the award in any person's name. You could use your, say, 50,000 Delta miles to send your friend from Amsterdam to Boston and back. (if they have an award seat, that is. Good luck with Delta!)

Don't barter or sell any awards! Certainly don't tell the airlines that she'd ever 'give' you anything for it! You are giving it to someone. Remember this.

Her being an international student does not cause any problems. If she has Visa issues with this or another country, and if these cause problems with the flight, that is beyond your control. She could lose the ticket if there's something wrong with her travel and residency status but assuming all of that is in order, she'll be fine.

You cannot earn miles for the flight if she flies but you do not. If you try to put in your FF number it will do nothing unless you both share the exact same name. I suppose if you were both named Kim Johnson or something, it could potentially work, but I am assuming that is not the case, nor would I recommend changing my name to a woman's unless that were my thang. It's not.

I hope this helped.

MM