Any automatic car wash is just a terrible idea if somebody is serious about their paint. Swirls are there from anything that has ever touched the paint, unless it was clean microfiber. There's a good chance the cleaner was Simple Green in some water - plenty of people use that to clean the engine bay for the same reasons it works on brake dust. It would also remove all wax, so the fact that the shop put a cleanser on the car then charged for "clear coat protection" (whatever that is) and "mild wax protection" (another mystery), THEN sealed it... wrong order. Also I like rain-x on my windshield, not so much on my paint.
The swirls can be removed with a polish. This can be done by hand, but somebody that knows what they are doing with a machine polisher can kick some serious swirl-ass. Remove swirls with polishing, then use a sealant, then - you gotta protect the surface with wax. Always always wax to protect the paint. The idea is that dust and contaminants get stuck on the wax - not in the surface of the paint. If there is no wax, anything that touches the surface can be causing swirls. Expect to pay a professional between $150 and $300 for a righteous detail - wash, machine polish, swirl remover, sealant and wax. It's expensive because to do it right - it takes most of the day. There can be several steps to machine polishing, depending on how bad the swirls are.
The "spider web" cracks concern me - would those be visible only on non-metal body panels? Spiderwebs are usually cracks in either the clear coat or the base coat. On non-metal panels like bumpers, it could be because of an impact - something hit the panel or the car hit something. On metal panels, I'd guess bad paint preparation at the factory or aftermarket paint job.