Honestly, I don't necessarily believe that a lifter needs to switch which hand is supinated and which hand is pronated while perfroming DL's and shrugs. If you're deadlifting any serious weight at all, your arms are already pointed straight down. I can't imagine how anyone could incorporate their bicep into their deadlifting while pulling anything over 225lbs. (and if you're not pulling over 225 you're certainly not overdeveloping anything)
Now I suppose it could have an effect on forearm development, but it shouldn't in any way effect trap or bicep development, but these effects would be very minimal on the forearms.
WRT to grip.. As Grizz already pointed out, you should be resting a few minutes in between heavy sets anyway.. if you're not, then you're probably not pulling very serious weight.. DL's take a lot out of an athlete, and it's necessary to have a little longer rest in between DL sets than probably any other exercise with the exception of squats.
A good way to build up strength is to throw your straps in the trash and just your raw strength to hold the bar, also incorporating pinch blocks is a good exercise for grip.. but honestly, if you're pulling heavy weight and doing heavy shruggs, then grip strength will come... it is a bi-product of pulling heavy weight from the floor or from racks.