Yes it's safe. Just remember to ground yourself before touching the parts. Just touch a metal object (the case will do just fine) before touching the hardware.
And you can leave the ram and cpu in the mobo, but I'd disconnect all of the other parts. You don't have to disconnect them, but I think that it's easier to do so.
And remember to put the small copper (I think) thingys under the mobo before putting it in the new case. So that the mobo is resting on the thingys instead of the case. If the mobo touches the case there is a good chance of "bye bye mobo".
Ya you want to use the stand-offs that come with your case, there will probably be directions with the case just read those and it will tell you everything you need to know about installing the motherboard in the case. If not your motherboard manual will have directions.
I've had really crappy cases in the past that are so cheap, they don't come with any type of instructions . The best part about transferring to another case is that you can clean all your parts and rearrange all your wiring and cabling.
*Fact: Microsoft Window's Blue Screen of Death vs Computerforums.org's White Screen of Death. Which is worse?