You won't have any air conditioning during the replacement so the answer depends on the weather and your tolerance for living without air conditioning.

Your duct work should all be sealed as part of the replacement work. I recommend testing for duct leakage after the replacement.

Make sure you have a lot of fans, since you wont have AC. It also depends upon your outside temperatures.

What about dust? Bugs? Thanks so much for your help!

Your HVAC replacement contractor should cover any openings between the crawl space and your home during the course of the work. Plastic sheeting and painters tape work well for that purpose.

Doubt, the dust and any bugs, will be any type of problem at all. I have a crawl space, and would not worry about something like that.

Expect it to be a noisy process though. :-)