what cheap parts could you recommend me for the vacuum gripper - suction cup and vacuum valve?
Is there some complete arduino solution for gently arduino controlled vacuum gripper?
How about something like this (click !) for the gripper.
You can find these kind of things at a lot of electronics parts sources by searching for "vacuum".
I'm not sure how well these will work on paper.
Or how well it will keep working after a while.
Depends on the paper as well of course.
These little air pumps will provide plenty of suction; mine has no problem lifting an 8x10 business envelope. The link isn't a great place to buy it but you'll find them on eBay, etc.
The pen gripper doesn't look like a good idea to me. Just use a suction cup from a nerf dart or something. The best cups will be silicone; the vinyl ones like are used for hanging Christmas lights on windows, etc. are a little stiff and get a "set" in them but they'd work. The ideal would be a "bellows suction cup" for which Google will provide plenty of examples.