You would inform anybody you choose to inform. You cannot expect that your past educational institutions will be particularly interested in a notice of a name change, and it's not clear what you hope to accomplish by sending them notice. Obviously, you would want to update your ID.
As for how long it takes.... the biggest part of that is you. You need to file your petition, you need to do that properly, you need to publish required notices, you need to make sure your case moves forward. If you want to know how quickly your local court can schedule a name change once you have all of the required documentation on file, you can ask the local court about how quickly they can get you on the docket.
Whether you would view yourself as having two first names or two middle names is a manner of perspective. If you want it to be more clear, you can consider having a hyphenated first name.

