I am worry about extreme stress around the trem pole slots and increased wear at the pivot points when tremolo goes down and the angle at the point of contact is very large in a situation where the strings removed.
For example, should I remove all strings, and then remove all springs? Or I should remove two central strings then remove one central spring etc. Other ideas?
It's not a problem with a little care and difficult to make any significant damage. Order of the springs is not important (and the claw will move more than the bridge will anyway).
Start with a folded bath towel under the guitar.
Then I place cards or post-it notes under the back of the bridge, detune and remove the strings. The spring tension keeps the bridge quite nicely in place against the mounting posts.
Flip the guitar over face down onto the towel, holding the bridge in position with one hand as I do so.
Remove the springs, and then flip the guitar over, again holding the bridge in place with one hand. Then you can pull the bridge back and up to remove it from the body. Be careful on guitars with piezo saddles, you risk breaking wires from the piezo to the control cavity!
When re-assembling, place the bridge into place and hold it gently into the correct position on the bridge mounting posts. Place cards/post-its under the bridge in the same position as before. Hold bridge in place while turning the guitar over. Re-install springs, string up as usual.
Never take off all strings unless necessary. Tuning is not the issue, yes it could be harder, but the real issue is that you are removing anywhere between 50kg and 70kg of tension on the neck depending on your string gauge. If you must take them all off, don't wait long before replacing them. Otherwise you may need to adjust the truss rod.
Nothing wrong with keeping all string on if you want but there is no danger or risk in taking them all off. It's an old myth that we need to kill off. I have never had to adjust truss rod after taking all the strings off. See myths number 3 and 4 here...
Truth About Truss Rods – Part 2 – Myth Busting! | DrKevGuitar.com