1) "That's going to be a pain in the ass to remember": the extensions are there so you don't have to.
2) "how do I login?": we're currently developing this side the service.
3) See 2)
4) We're planning on proposing mobile apps too.
To put it into context, nBox is fairly new and we did not implement all our ideas before knowing if people were gonna be interested.
In regards to the extension though; that's just such a bad direction to go IMHO. Most email is being read on phones these days anyway; does NBox even work from a phone today?
It works on android phones with chrome, but like you said I wouldn't remember the generated addresses, so we think the extension is a big part of the service, and unfortunately, extensions are not available on mobile. We're working on the subject to find the best solution possible.
A password manager generates random data for the password textbox. Your extension generates random data for the email textbox. Perfect!