I second what's been said so far- esp. regarding respect and about being available to the crew. Once you develop a good working relationship with a crew, they can be really helpful to you. (And if you don't have a good relationship with you, they can make your job really difficult). If you can find a way to either make their job a little easier, or do something nice for them, it really goes a long way. My fellow ASM and I used to buy the props crew those little Keurig coffee packs for opening night- they loved them and the theatre only bought loose ground coffee. Candy is also pretty popular.

The crew will do spikes, but I would advise keeping a list of spikes and checking them every night. Even if the crew checks them, you never know if one will get pulled up by accident. I did an show in a large venue where the set was struck to make room for a one-night concert and loaded back in the next night. When it was loaded back in, one of the floor panels was flipped the wrong way- even though the crew checked the spikes, they ended up out of place.
It's good to figure out what tasks fall to which department in a particular theatre. In my experience props usually handles moving music stands and step units for assistant conductors to use for conducting coro or banda backstage, but deck elex might be responsible for stand lights and moving roving monitors into position. It's good to check and make sure what the breakdown of responsibilities are at your particular theatre so that you can break down changes accordingly.
Definitely agree with the diagrams- I am not particularly skilled with making little clipart prop diagrams (though I know other SMs who are) so sometimes I end up taking photos of a complicated set up and label accordingly.
Make sure you date the paperwork and possibly give it a version letter. I had my guys take notes on their paperwork and then took it back at the end of the day to make their corrections and make sure old version weren't floating.
YES. I've also done photocopies on colored paper (in addition to dating and labeling) depending on what version it is. For instance yellow might be for piano tech, green for piano dress, pink for orchestra tech, blue for orchestra dress, and white for the final version. Makes it really easy to see that a crew person has not yet received the most up-to-date running paperwork. (I have worked at companies that have standardized colors for particular rehearsals- worth asking the SM or TD if there's a preferred way to do it).
What BayAreaSM said about you knowing the show is absolutely true- a lot of what I end up doing backstage at IA houses is double checking things. That all the props are set correctly, that the crew heard the SM give the 5 minute call on the next scene change if they don't have a headset, that the crew is on their way when their next handoff or quickchange approaches, etc.
In this case, you are truly stage management - you make sure everyone is doing their task at the right time and that everything is running smoothly.
Well said.