Did you see em do the work?
Looks like they didnt do a good job of it..

Your first pic looks like either clear coat degraded and flaking off (polish cannot resolve), or calcium material harden by parking in carpark basement and water leak from ceiling hit onto car (need to use chemical to break the calcium material before polishing).

Your second pic scratches looks like too deep, maybe cannot remove with polish or compound anymore. One spot at the bottom section definitely looks like it has paint scratched off dy.

Your third pic water spots can be removed, but just polishing alone wont be enough, usually needs to be compounded.

Maybe u wanna get second detailer opinion.

On your query for #3, compound is meant "heavy polishing" in layman. This is achieve by using more "stonger" polish.

Compound that you could find and one of the best for consumer is Meguiars Ultimate Compound. This is available also at lazada, sold by Meguairs Malaysia. Buying online is much cheaper than shop and free shipping too. You can get using discount on www.lalalaxx.com.

http://www.lazada.com.my/meguiars-g17216-u...nd-1445740.html

For your issue 1, I could recommend wet sanding with 1500 grit follow by 2500 grit sandpaper, then compounding and polishing. However, I have to say first that there is no guarantee this would completely remove the marks as just from your photo alone I cannot 100% say what is the actual issue with your paint.

As mot88 said, compounding basically is a more aggressive form of polish. Compounding is meant to remove or reduce your paint defects such as moderate scratches, water spots, etc. which usually cant be removed by polishing. Shops may or may not tell u whether they compound your car or not because it is a detailing technical term, so instead they will tell u how many layers they polish.

Mmm, I think have to be realistic that our weather conditions will always make car dirty especially during rain and shine and rain seasons. To maintain some shine it is always good to wash the car on weekly or 2 weeks basis.

Polishing is not necessary or recommended to be done weekly because it will thin your paint. People only polish when necessary to remove defects and restore shine (usually 6 months to 1-2 years once).

Waxing usually is done monthly to few months once depending on wax durability, severity of weather and your washing consistency. Waxing helps to maintain shine and reduce contaminants bonding with paint. U can always use spray wax after wash as quick fix to maintain car shine.

Even if u got coating applied from any shop, washing regularly is still recommended to maintain shine.