I use an inkjet printer and just use standard ink-jet paper. The cover insert on a DVD case need to be a little bit smaller than 11 inches. So a standard 8-1/2 x 11 sheet *should* be fine. My printer has .25 inch margins and on some cases it is just a wee bit smaller than I would like (maybe .25 inches), but I can live with that.
If you have a "multi" DVD case with a wide spine you may need to use a 8-1/2 x 14 sheet.
It's inkjet paper, so yes. But, photopaper is relatively expensive and would be a waste of money in my opinion. Your DVD inserts do not need to be that high of quality to look good.
Plus, that paper will be fairly thick - like a photo print. And it will not fold very well so the spine of the case will probably look bad. You want a fairly thin/light sheet of paper for a DVD insert.
So what paper i should buy? I buyed normal 80g paper with my printer, but when printing with 'normal' quality, paper gets too "wet". With 'fast' quality paper is not so wet, but quality isn't that good.
You should try some "regular" ink-jet paper. Standard paper is not meant to absorb all that moisture. Still, when using high-quality the output may be a little wet. Just give it some time to dry.
The best paper I have found for doing my DVD covers and inserts is from Office Depot. It is their color inkjet paper 24 lb weight and 115 brightness. It runs about $4.99 a ream. I've used it on about 2500 covers and like the thickness, brightness, and best of all the price. Their SKU # is 562903.