I did say "most" states. I used to work at a car dealer's service department, and I've seen some cars come in where they don't have a state inspection sticker because the owner just moved from a state that doesn't do state inspections.
Because of that job, I've also seen stickers from other states that do do state inspections. Most of them will be on the windshield either in the bottom corner on the driver's side or in the bottom center. They'll list a month and year as numbers (ex. 8 and 24 for August 2024) for when the inspection sticker expires.
So, at least in my state, someone with 8/24 on their sticker would at least need to drop their vehicle off at a mechanic (not necessarily the car dealer, just a mechanic authorized to do state inspections) by the last day of the month. If the mechanic doesn't do the inspection until the the beginning of the next month, that's fine. You'd just get a sticker that says 9/25 on it when they're done.
The state inspection in my state costs a set, predetermined amount. The mechanic legally cannot charge more than that amount. (I think it's about $10, but I could be misremembering. It's somewhere around there, though.) If anything doesn't pass inspection, that's where the cost can come in. How much it costs depends on where you get the thing fixed.
In my state, if you car doesn't pass inspection and you choose not to get the thing repaired right away so it does pass (maybe you don't have the money right away, or maybe you want to bring it someplace cheaper to get it fixed), you're given 30 days to fix it, regardless of the original expiration date in the sticker. They replace the inspection sticker with a special "state inspection rejection" sticker that says by what date the inspection is due. At my old job, I used to see plenty of people come in on the last day of the month knowing their car would fail inspection. They came in to get that 30 day extension.
Generally, inspections don't take too long to do, though that can vary depending on where you bring your car. They usually won't take longer than 24 hours, though. For me, they're usually done either within a couple hours of bringing the car in (if I bring it in early enough) or by mid-morning the next day (if I bring it in in the later afternoon or evening). So I could bring my car in after work on the day before my day off, and it's pretty much guaranteed that it'll be ready by about 10am the next day the next day (or end of day if they find something wrong, depending on what it is).
When I was working at the car dealer, customers waiting in the lobby would typically have their car ready within a couple hours, and customers who had alternative transportation would usually have them done by the end of the day with enough time for us to call the customer and for the customer to come by and pick up their car.
This definitely isn't the case everywhere, but the dealer I worked at also gave out free loaner cars for customers to use depending on the circumstances (ex. did the customer have an appointment) and availability (ex. do we have any loaner cars available right now). Customers just coming in for a state inspection wouldn't typically get one, but, if there was something that didn't pass inspection, then we might set them up with one if we had one available and if the customer wasn't able to wait any longer.
Tl;dr: In my state, getting a state inspection is usually relatively painless. The inspection itself doesn't cost much, but the cost of fixing something so it passes inspection depends on where you go. There are also ways to get extensions if you need them, and, if you plan ahead, you can usually get your state inspection done on your day off without any problems. Even if there's something that'd cause it to fail inspection, it's usually still done the same day.