Upvote:2
Your question as to where to apply (LA) is answered on the Czech Embassy website: http://www.mzv.cz/washington/en/consular_information/contacts/offices_in_usa.html
Your question about length of stay would be best answered by calling the same consulate you would apply at. They would know best what visa you need (if any) and what supporting documents you will need. And pray they don't tell you to get a student visa as that can take two months to be issued. :(