As long as you're going to drink part of the beer pretty quickly, it shouldn't be a problem. I usually put my hops into a muslin bag and wedge it between the dip tube and the sidewall about 1/4 way down the keg. You don't want to put it too far down because if the beer sits on the hops for too long, it will leave you with grassy flavors.
Do it all the time. Stick the hops in a 1-gallon paint strainer bag, tie it off with unflavored, unwaxed dental floss and route the slack of the floss between the lid o-ring and lip of the keg (that shouldn't affect the seal). Then taste frequently and pull the bag out once the flavor and aroma are where you want them.
Yeah, keep 'em in a bag and keep the bag from getting into your dip tube and you should be fine. I also find that in the cold, the grassy flavors take a lot longer to develop. I don't think I'd let them sit on the hops for a year, but a few months shouldn't be a problem. (Of course, different hops are liable to give off grassy flavors differently, so YMMV.)
In the keg? They don't- they creep up and clog up your diptube if given a chance!
I use either a hops bag or a "tea strainer" ball for dryhopping in the keg. I use a tightly woven bag (sanitized) for pellets, and the tea ball for leaf hops. I don't tie them to anything just drop the bag/ball in. Cold temperatures take longer to dryhop, and I've never had grassy flavors as a result as dryhopping in the keg. I never remove them, and I've had good results.