If i tune my guitar with a FR to drop D, will I have to readjust the springs?
YES, but just a little bit.
Maybe...but likely not. If the bridge comes up too much for you then you'll have to adjust it...just depends on your current setup and what string gauges you're using. You should be fine though.
I had to with my Ibanez (it could just be the edge 3), just take your time with it and if you do need to ajust the springd dont turn them that much.
Ideally yes, although to be honest you'll get away with it if it's just dropping the low E. You'll still need to re-tune all the other strings though.