Welcome!
I agree with Daisyl. "I can't believe that a landscaper that was any good would not guarantee their plants and their own ability to plant them properly. " And I would add, 'especially at that price'. I know that sounds cynical but it sounds like you're having some major landscaping done.
If it's not too late, I would certainly get some other proposals. And if I did nothing else, I would at least ask around to see if it's true that "no one around guarantees or replace plants". That should just take a few phone calls. I do know that if I buy one plant, I'm unlikely to get a guarantee. But you're talking about way more than that.
As for planting schedule, I can't advise you much other than go with your gut. Maybe see if there's a Master Gardeners' Hotline. (We have one and I'm not shy about calling them with questions.) Or there's bound to be some local resources to talk to, like maybe a gardening club or even an arboretum. Or what the heck? If you call other landscapers for bids or to ask about guarantees, might as well ask them about the best time to plant. Just say that you're curious.
I'm in Zone 8a and wouldn't dream of planting shrubs and trees in August. My understanding is that Fall is THE best time. Our problem in Texas is deciding when it's actually Fall.
I'm going to see if I can get you a good guide but maybe someone else will have already answered before I edit this.
Edit: We didn't ask you what shrubs and trees are going to be planted. It's important to make sure that the plan has an evaluation of your soil, existing sun/shade, whether the plant is considered 'invasive' (don't believe it's ok because everyone else here grows it) and so many other things.
Anyway, here's a resource to play around with. It can be a little overwhelming, at least it was for me. But just play with it and see what you can find out.
https://plants.usda.gov/java/
I also did a quick google about the best time to plant trees and shrubs in northern NJ. Every single site I looked at said that fall is the best time. Not one site would agree that planting in August is ok.
Ok. I'm getting obsessed so I'll just add one more question. Do you know if your landscaper is a certified arborist or has any on staff?