Yes to everything Sherrie said.
I like the cake comb idea for cutting in front of people, a gloved hand gets messy and just isn’t nice to look at. The knife gets messy, too, so have something ready to deal with that.
I’ve never encountered a need to turn the cake on its side to prevent squishing out of the filling, I think that happens more when you push straight down without any sawing motion, and/or if you don’t have serrations on your knife. Maybe that’s more of a concern with a warm, outdoor event, which I haven’t done.
Yes to removing the tiers for cutting. It’s possible to carefully cut the tiers while resting on their straw supports, but if the straws that support underneath aren’t close enough to the edge of the tier that it is supporting, the tier will tip over when there are only a few pieces on one side (I caught it when this happened, but yikes, it was stressful!).