I have now got (almost) all the paperwork I need for my ESA application

Probably something you're well aware of, but absolutely make sure you keep copies (or only send copies) of everything. DWP have a bad habit of losing paperwork and then claiming it was never received. Plus you may end up needing it for other stuff.

having spent 10 weeks in California in 2011 on a summer placement apparently means that I need a statement from the Home Office about my residency status

*headdesk*