I'm worried that taking my strings of every time to check whether the action or intonation is right will make them snap. So what's the safest way to remove new strings so I can change the intonation and action.
Why do you want to take the strings off to do that?
Slacken the one specific string you are working on slightly if you are raising the action or adjusting the intonation rearwards; keep compensating for pitch as you adjust.
Otherwise it's just repetetive guesswork as to what it will be like once you reassemble it.
With a Fender style bridge you just need to loosen one string a little and adjust height and intonation, then retension.
To adjust intonation on a one piece (Les Paul style?) bridge you need to loosen a string quite a lot more to allow access to its intonation adjustment at the rear of the bridge.
For height adjustment on a one piece bridge you'll need to loosen all of the strings. There seem to be quite a lot of ways to adjust the height of these bridge types, so it's best to search for info or ask about your particular model.
Don't worry too much about string damage - in my experience loosening the strings but keeping them attached at both ends will only do minimal damage as long as you're careful. They usually break at the machinehead end, if you try to remove them completely.
There's really no need for that. For intonation fine tuning, the little screw can be adjusted while the string is under tension, otherwise, the string will need re-tuning each time. As far as action goes, again, there's no need. In fact, it gets in the way to de-tension. A Tune-o-matic bridge has two screws, which will turn while the guitar is in tune. A bridge with individual saddles can also be adjusted under tension. You may be confusing this with truss rod adjustment, which is kinder to the guitar when the strings are not under tension.