I haven't worked an opera with an IA crew, but the ballet company I work for has always used IA.
Bwoodbury has given a lot of good advice - and here are a few more things I think can be helpful:
1. Always be available. You are more familiar with the show than the crew. They are going to have questions and you need to be easily accessible. My previous ASM used to wear brightly colored shirts to tech so that the crew could spot her. (She would switch to blacks for Dress and Performances.)
2. Know where you are in the show. There have been times when a crew member wants to know how long it is to their next cue (they need to fix a problem, need a bathroom break, etc) and the SM may be too busy calling to give an answer. I personally time stamp my run sheets by the activity onstage, so that I can follow along and let someone know how long they have until a shift or a hand off.
3. Respect your crew. When I was interning in a major Lort house I learned that the crew had no respect for my fellow SM intern who was working the shows opposite the ones I was on. When I had the opportunity to watch her interact with the crew she was incredibly casual, giving people nicknames, being very touchy and acting generally ditzy/giggly. When you're the new kid in town, you need to act professionally and respect your crew. You have to give respect to get respect. It took me some time to get comfortable with my IA crew, and while we may joke around, I am still respected by my crew and my directions are followed.
Bwoodbury's advice regarding paperwork is probably better than mine. I would normally generate separate run sheets for Props, Carps, and Rail/Winch. I would then leave it to the appropriate department heads to assign specific tasks. Your ASM Run Sheet should include all of those items. For Ballet, I also include any Deck Elec tasks which involve shifting rovers for set changes and wardrobe receives/hand offs - all with timestamps. At the ballet it's the wardrobe supervisor's job to generate the notes for the dressers, but I needed to be aware of their tasks - as sometimes I had to warn them when their receives were coming up.
The one thing that can be tricky when working with IA is when to step in without stepping on toes. As your run sheet should have all of the tasks for the various crew departments, you are aware of when and where things are happening. One thing I have noticed most often is a missed prop receive. Usually I will stand near where the receive is going to happen and wait for the crewman to show up. Most times they do walk up right before it is supposed to happen (as they are following their paperwork/notes as well) and there are no worries. However, if the cue is coming up and they are reading a book, I would go over to them and let them know that it's time for their cue. After doing that once, they tend to not miss again.
There have been instances where I was waiting for the crewman to arrive and I could not find them so I simply stood in their place, received the props, and did not move. When the crewman arrived he took the props from me. That particular moment was an important receive, as the actors were running off into a wing and I knew they had to run through the cross over for a quick change, so they couldn't wait for someone to take their props (and could not run with them either). The crewman cannot be mad at you if you couldn't find them and it's their job to be there. I felt the important point in this was that I did not move once I received the props. I wasn't trying to do their job (I didn't put the props away), I was merely standing there and someone gave their props to me.
One other item that can be touchy - giving a Clear to the SM to indicate a completed shift. In our theater, that is the job of the Head Carpenter. I had never been told this (when I was the ASM), so I would be give the PSM the Clears to move on. It actually took a few years for the Head Carp to finally say something to me, and I apologized for my lack of knowledge. ASMs still give Clears to let me know the talent is set (places, bows, etc) but my Head Carp is the one to tell me that shifts are complete.
And you are correct: don't touch props, don't touch costumes, don't touch anything that isn't your run sheet or your pencil. If you notice something is wrong, find the appropriate crew member to handle it. If you need a dresser because someone's costume is coming undone, find them or have them paged. The talent should not be dressing each other, nor should they be doing each other's hair (assuming you have union Hair and Makeup). Spikes are generally set by the department they affect: carps do their own scenic spikes, elecs spike their rovers, props spikes the props - and in the ballet, they generally spike for the dancers, if needed. 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.