What kind of glue to use?

I would look into using glue dots. Many craft stores carry them in their scrapbooking sections. It works similar to white out tape- it's a roll of sticky double sided tape. I've used it many times for projects and it comes out seamlessly.

Hi Kim

These look awesome! Love the key!

I would agree with the previous poster, glue dots may works best!

Perhaps test some out and see if you get the desired effect before doing all of them

How many place cards do you have to make?

I agree with testing out various materials before you use it on all of them. I would personally suggest classic glue sticks. Unless you put a whole lot on the back of the paper, it shouldn't curl up. Cover the outline of the paper in the glue stick, plus parts of the center so it doesn't bubble up, and make sure you smooth it out when you place it on the cardboard, but it should work beautifully. It'll give you a smoother edge that's less likely to come up than glue dots will.