I got a coach almost right away when I started racing. I found having one very helpful in a lot of ways. One of the things that stood out was when I told him I didn't think I'd every be a crit guy. He disagreed, which actually turned out to be right.
I'd add in "age appropriate experience" to Hermes list. You don't relate to a 20 y/o like a 50 y/o from a training/life/racing/health perspective.
Also:
-Ask about communication. How, how often, topics, Etc. Especially how often you'll get feedback.
-How far ahead he/she schedules. If you're getting a month or two of daily plans at a time you're not being well served, fitness, success or failures during workouts should all be part of developing the plan and you can't forecast out where someone will be in a month. That's not to say you might not have an overall plan (base/build, Etc) that might forecast out, and if you're periodizing and have an "A" race you should have such a plan.
And what Kwaki wrote also has a lot of wisdom. Mentally this sport can be brutal.