You should be including this information in your ECAD data. It goes on the manufacturing layer, along with the other information such as dimensions, stack-up, and special features.
The surface finish solely determines manufacturability, at least at the level I work at. HDI or RF might be different.
HASL is generally the cheapest option, and it is suitable only for low-cost, high feature size PCBs. It is vaguely flat, has very few issues with storage or multiple reflows and is fine for 0805 size components or bigger, TQFP at a pinch. DFN/BGA is a really bad idea on HASL.
Immersion Tin, the next step up from HASL. It has good solderability, but storage can be a pain. Multiple reflow cycles can be a pain. Better than HASL if you treat it correctly.
Immersion Silver has excellent solderability, but storage/process is a serious issue because of oxidation. It is cheaper than gold finishes, but that is the only real selling point over them.
ENEPIG is an improved version of ENIG basically. If you are doing high-volume then you should look into the difference, otherwise these are the best all-round surface finish. Flat with excellent solderability, storage and multiple reflows are not a problem.
Organic Coatings (OSP etc...). These are very flat as they are just a thin film applied over the copper. Solderability is excellent, but if they get damaged, and even slight rubbing is enough, then you get oxidation and it all turns to worms. It can have issues with multiple reflow cycles.
Other finishes are more special purpose (hard gold for edge connectors etc...), you board house would likely be very happy to chat with you about the appropriate finish if you have questions about the best one for your particular application.
The finishes are only applied to the outer layers (usually anyway).