Is it safe to run/jog with a Flash drive Macbook Air?
There are definitely no moving parts. I wouldn't worry about jogging with it. I wouldn't even worry about jogging with a non-flash drive equipped laptop if it was turned off. The drive spindles and heads are immobile when the drive is powered down.

For a pack, look for something that suspends the laptop away from the sides. Preferably holding it in something that'll absorb the shock a bit.