Pierre, if picks slide away from you, you're not the first one. Try asking for Tortex, or the nylon ones with a grip. Also just keep practicing, it will happen lessas you get more accustomed to using the pick.

To go back to one of your questions: YES, you need to use a flatpick to play a mandolin effectively. It's just the way it is, your guitar fingerpicking is not relevant to this instrument, and the rest of us are not Ry Cooder.

Personally, after 30 years of mandolin picking, I am exactly in line with The Levellers recommendation--I use Jim Dunlop .73 mm tortex picks almost exclusively for both mandolin and guitar. Because I pick fairly aggressively, the points wear out fairly quickly on mandolin. I find it convenient to use the same kind of pick on both instruments, but would maybe go one thickness heavier (.88 mm) for mandolin only.

I use regular teardrop shaped picks--the big triangular ones are too floppy and cumbersome, and I just don't find any advantage in the small tiny ones.   

As you're starting out you may prefer a lighter gauge pick. It flexes a bit when you strum it, making it easier to hang onto. (A heavy pick digs into the strings and is more likely to jump out of your hand, because it doesn't flex.) However if you find yourself learning to play faster (bluegrass or Celtic styles), the flexing slows you down because the pick point is boinging around, making it harder to place when you only have a fraction of a second between notes.

Good luck, try different types, see what works for you.
W-O