Yes, they can be placed for access from outside as well as inside the shower, inside the opening.

Yes, but if they are OUTSIDE, ask the GC to install a second set inside for adjustments during the shower. If you are having a curbless shower, watch the drop of the shower floor and the placement of the drain(s). Especially tricky with 2 shower heads. Looking at the specs, not having a door may be a problem if you need a large opening relative to the shielded portion.

I don't think you are allowed to have any controls Outside. Just accessible to the outside.

"So, my question is, is it feasible to have the shower controls so
they're reachable from outside the shower (on the wall between the
shower and the vanity)?"

Yes, you could place the controls on the 2'3" wall. If you place the controls there and the door will be hinged on the left (shower/closet wall), if interference between the shower door swing and the vanity could be a factor, consider using hinges that allow the door to swing both in and out of the shower. That way if a vanity drawer opens up, the shower door can swing in and you can egress the shower.

"If we can manage it, we'd even prefer not to have a door"

If you go doorless, with the width of that shower you'll likely get overspray or rebound splash on the bathroom floor. Because of that, you should extend the shower membrane out under the first few feet of bathroom floor tile as well. Have your installer use a capillary break at the shower threshold so moisture does not migrate from the shower floor tile's thinset and grout out into the bathroom floor tile thinset and grout.

If you decide to go without a door make sure the design allows you to add one later. Many people find showers to be drafty and cold without a door. It just depends on the air flow in your bathroom.