No, there isn't always a women's race, and usually there's only one for all cats of women. For your first crit or road race, I'd look for a larger race that specifically says the cat 4 women have their own start time.

You can race down one Cat with the men, so as a 4 you can race the men's cat 5. Since they're such bad bike handlers, it's not the best idea if you have other options, but depending where you are you may be forced to race with guys.

You don't have to be on a team, but since you're from a tri background, it is imperative that you learn to ride in a tight group, bump elbows and wheels, etc before attempting a crit or RR. You don't want to be the one in the pack that is obviously unsteady. TT is another story-no need for group experience.