Well, that's quite a list...
Yes to most of those questions...
I think it depends on your experience (years in the OR) and confidence...
There's always the "turnover" rush...when I worked at a major trauma center we had to scavenge for equipment like headlights, beanbags...people would steal from your room if you were not there...sabotage big time. I'm currently at a quickly expanding community hospital. Our turnovers are always 15 minutes or less, we have excellent ancillary staff (housekeeping,anesthesia techs) who care about what they do, and enough equipment so sabotage does not occur. So now, I'm not so stressed about the next case. If I were working days (not anymore), I would get to work early and get my cases situated (extra equipment, sets, suture, meds)...thus less stress during turnover.
Counting...try to do it before I bring a pt in. Not always possible due to that rush for turnover. We absolutely do a count before the surgeon gets the scalpel...same for our timeout. I don't care if it's a blase...this is john doe and we're doing a lap appy...it still works. You should always count for cavity cases...big cavities...like the belly/chest you should always do an instrument count. It also depends on the scrub nurse I'm working with. Some like to get set up and then break until the pt is ready. I'm cool with that because I'm not running like the proverbial chicken trying to deal with the pt, anesthesia...and now my scrub nurse. Others like to lounge around and scrub in at the last minute and then ask for various items they could have gotten if they had taken the time to prepare. I don't mind getting items but piss poor planning on your part does not make it my emergency.
Yes, I have anesthesia give suture when I'm out of the room. Usually they're kind enough to tell me or write it down on the grease board. If not, my scrub nurse is usually kind enough to let me know. They can be quite helpful if you're out of the room...they can also be a pain.
We always do a time out. That wasn't always the case but even before this was JACHO doctrine if it was an extremity/eye/hernia/etc I would always time out to verify site/procedure.
Again, there's always a rush to get the case started. Most anesthesia providers/charge nurses are cool and willing to give you an extra minute to catch your breath. Of course there are exceptions to the rule.