There is no deed on a co-op. When you own a co-op you own shares of a corporation and you get a proprietary lease. Every co-op has their own rules about what they want to see in terms of assets and liquidity after purchase. And lenders have their standards which are tightening up. If your preference as mentioned is to co-purchase, that does make your daughter a stronger candidate both to the seller and the co-op board because your financials, which I assume are stronger than your daughter's, will give an overall more financially sound picture.

Yes, if your name is on the deed as joint owner with right of survivorship, you are equal co-owners. Upon the death of one, it passes by law to the other.
If you gift 20%, she can borrow 80% but I am unable to advise of rate or terms.

Is this for a co-op or condo?

Call your CPA, your attorney and your mortgage broker; these are the people who are appropriate to answer this question.

Check with a real estate lawyer. I'm not a real estate lawyer, so this isn't legal advice. However...

I don't see any reason it wouldn't work. The drawback to being a co-owner is that your name would be on the mortgage along with hers. In addition to making you responsible if she defaulted, it would knock a few points off your credit score. But that sort of arrangement is done all the time. In fact, the first house my wife and I bought, my mother co-signed and is a co-owner.

You might also look into equity sharing. That could give you a lot of flexibility in how you can help your daughter.

To figure out how much of a mortgage she'd be able to qualify for, you'd have to talk to a good mortgage broker. A major variable, I'd guess, in addition to her credit would be her employment history and employment prospects.

Good luck.

She really needs to see a lender and get prequalified.