1.) Best way to industrialize?
2.) Best way to keep my new Korean acquisition?
3.) How to deal diplomatically with the US / how to get allies against/with other European powers?
1) I always start off with a cement factory, I believe in the southern-most state. In not too long you'll also want to research the tech that lets you build machine tools factories. You might also build a single lumber mill in your northern-most state, since it has timber but not many people.
Liquor factories usually do well, and the glass factories that feed into them. Don't bother with fabric, since the UK will probably be monopolizing the dye supply. Don't over-produce small arms , but artillery and explosives will have decent demand provided you build enough artillery; ammunition also needs to be produced a lot. A furniture factory or two wouldn't be a bad idea, but check the supply of fabric before you go for clothes. Don't over-produce steel, since it uses coal which you need a lot of, but you will need a fair amount.
2) Eh, mostly raw resource extraction. You can stick some factories up there, since all of your clerks and craftsmen pay taxes, so just do what works.
3) Wait until 1860 to get your navy going. Research the level-three naval base and start building those before, so that when 1860 rolls around you can get right to Iron Steamers. Build a whole bunch of ironclads and commerce raiders (I always went with a 1:1 ratio) and the AI can never touch you. The Strait of Malacca is the most common choke point you'll be using. It will funnel their fleets right to yours, and you'll win every battle. I never bothered taking the Philippines from Spain since they're worthless, but Malacca, Java, and Sumatra are all good investments since they'll get rubber, and Borneo gets oil.