Generally, with a guitar in good condition, no. The only time I've detuned a guitar was when shipping one via air, where it might be in an unpressurized baggage compartment. I think the strings should be kept at pitch - but you should check and adjust them if they are getting sharp (which can happen with temperature and humidity changes.)
If you play a guitar that is in good condition regularly, you should notice any issues as they develop and address them before they get too serious. I would think once a week would be often enough to adjust the tuning and notice if any problems were developing.
I suppose that if you are storing it for long periods and you aren't playing it regularly, detuning might be a reasonable precaution. I would think in that case that tuning the strings down as much as one-whole step might be reasonable, but I wouldn't leave them totally slack.
I don't think you need to worry at all about the nut. That should be a non-issue.
EDIT: On the other hand, if there is a structural issue with the guitar, it may well make sense to detune the guitar and keep it that way - until you get it repaired but detuning the strings is not a good alternative to getting the guitar repaired. In your specific example, the bridge could still come off completely when you tune up the strings and possibly injure you or someone else, and cause more damage to the guitar.