Trying to help someone rebuild their credit

My fiance (30 yr old) has crappy credit - CreditKarma score of 569 (very, very bad, I know - she made a lot of stupid mistakes when she was younger). She obviously does not have any credit cards, but I want her to be able to get some of her own in the future. I'm trying to do some planning for our future, and would like her credit to be improving by the time we start house shopping (though I expect to have to bare the burden myself).

Things going for her:
1. Completely paid off a vehicle.
2. Completely paid off one student loan (might have had some missed payments in the history).
3. Pays cell phone bill monthly and on-time for several years.

Things going against her:
1. Still one open student loan (might have had some missed payments in the history).
2. Had bad debt/accounts in collections for a couple of years due to an insurance issue (and some bad decisions - they were for less than $100 but she didn't pay them on principal until I made her). I know this, more than anything, is negatively affecting her score.

I first ran her free credit score through one of the agencies when I wanted to start the process of rebuilding her credit and discovered the accounts in collection. We have since paid them off (used the last free credit score to verify), although one of the collections agencies still try telling her she owes them money. I can't get a credit report to verify that, nor have they mailed her anything proving. I'll consider it closed until I see otherwise.

My plan:
1. Pay off the student load in full (my thoughts on this below).
2. Get her set up with a Secured Credit Card from my local Credit Union (no rewards with it, but no annual fee either with an 8.88% APR).
3. Have her pay off the student loan and her cell phone with the Secured card (not certain if her student loan company, ACS, accepts credit card payments).

My questions:
1. It hurts having her pay for things on a card without a bonus. I have pretty a pretty good credit score and have been setting up my cards to maximize my rewards. If I add her to one of my cards (say, my Amex Blue Cash Preferred), would that help her at all? Or since the responsibility to pay would be entirely on me, it will affect only me? If so, how would it affect me (assuming it's paid on time and not maxed out)? I've never added anybody to one of my accounts before, so I'm not certain how it works. They ask for her social, so I don't know if they do a hard credit check on her.
2. Would if be better if I paid off her student loan with one of my cards (1% cashback max with any of my cards) to clear it off ASAP? Or would it be best to let her clean it off on a secured card (assuming they accept cc payments and it would need to be done over a couple of months due to secured card limits).

[Edit: Added question]
Additional unrelated question: I have a card that has 6% cash back at grocery stores. Is it correct that I can purchase gift cards, and receive the 6% back on them?


Any and all suggestions, advice and help are greatly appreciated.

Thanks in advance.