If you have get back to your strings really quickly after flipping a lever, it is not necessary to go all the way back to the middle of the string. Place wherever you can get to in the time available, and work your way down as you continue to play. Practice your lever flips like choreography, in time. Practice them without actually playing any music, but just placing your fingers and then going to the next flip. If you haven’t already done so, put red and blue color of some sort on the F and C levers. (Some paint nail polish on the levers, others use hair elastics. There are some specially-made colored widgets that slip onto levers as well.)
As for hand position, experiment and see what works for you. I just grab the lever between thumb and forefinger, but technique may vary depending on the type of lever.