@Michael Johnson There is A LOT of paperwork involved in getting the unit accepted for a CHA voucher (in fact this is one of the downsides that hasn't been mentioned, but one small error in the paperwork jeopardizes the timeline for getting a tenant to move in) but one area you have to fill in is the "requested rent amount". I always swing for the fences when I fill that in, knowing that even if I fall short it will still (very likely) be approved at a market rate. Oddly enough, they have approved my big ask each time I've brought in a new tenant or requested a rent increase for an existing tenant.
I can't speak to how long getting the building approved by CHA takes, as mine was already approved when I had purchased it, but the individual unit inspection typically takes less than an hour. Depending on whether or not you pass, the follow up inspection might be a week later depending on how busy their inspectors are. On my most recent inspection I was failed for not having the unit number on the entry door in the common stairwell... yes, they will fail you for something that simple. Now, if I had a #2 sticker laying around I could have slapped it on the door right then and there and they would have been fine with it, but it was a lesson learned for me. Other simple things include grounding/reversed polarity of outlets (you can buy an outlet tester at Home Depot for $5), using foil-faced tape at furnace & water heater flue pipe connections, operable smoke/CO detectors, cracked/chipping/peeling paint if your applicant has children under the age of 6 (I think?), among others. It really just depends on how the inspector is feeling that day, and how many more inspections they have scheduled the rest of the day.