Most printers will take quite a load. I have even done sheets of magnets!!
I buy photo paper at Costco and it works well. It's a little sturdier than most cardstock. You can get glossy or matte I think, but the glossy is so nice. Make sure when you print, that you print on high quality, and you will be happy with the results!
Yup, I would just print out the photos on the Kodak stuff, cut them, and then put them onto the card. You could do 4X5, or 5X7, but the larger one would probably cost more to mail. Go for it. Try one and see what you think. Or print out one page of pictures and see what you think. Print the pictures first on a plain piece of paper so you can see what it looks like and it it's what you want. Just keep tweaking until is!