Re: How much?....
Taking all the strings off isn't a great idea if you don't know what you are doing but I've done it many times when building my guitars and/or replacing pickups. The reason you don't want to do it regularly or a lot is because it can stress the neck/truss rod. If you are using the same gauge strings, the trem should be very close to the same position as it originally was once the strings are tuned. If the trem is higher, that means one of 2 things:
1) you installed thicker gauge strings
2) possibly a tremolo spring on the back came loose?
If you are trying to use thicker strings you will need to either tighten the claw screws on the back or possibly add another spring.
If there are 3 springs on the back and the guitar is in tune with the too high bridge, just screw the two screws into the guitar a little farther and recheck the bridge height. Once you do that, the strings will need to be slackened some because they will have gone sharp so retune them and that will make the bridge come down some more. You will likely have to do this process a few times to get it balanced. If you find you are screwing the screws in almost to their limit, it's time to back them off and add another spring.