If you are only staying for a week then you are probably trying to do too much. Cornwall isn't that far in miles from Liverpool, but it is still a heck of a trek by car to get there and virtually a full day by train. Given you are in Liverpool would the Lake District not be a better option? It is a sight nearer and easier to get to.
I live in Leeds and as a born and bred 'Loiner' would defend my city to the hilt, but would confess it's not tourist central, unless there's something specific here that has attracted your attention. It rewards exploration but not the kind you might be seeking during a week in the UK. I'd say something similar about Manchester.
York yes, London, of course, Bath yes, Cambridge....maybe. Cornwall I'd save for another trip and do it justice! And to do that you'd need a car - it's bigger and more remote in terms of infastructure than you might think.
B&B's I'd do internet searches on.... they are always a bit pot luck, but most are OK, after all they are businesses in the end and need repeat business. That said, I've stayed in a couple of pretty crummy ones, but these are far outweighed by the exceptionally good ones. Have a great trip and I hope you smash the half!
Ian