Hi Alison -
Earlier poster is correct - this is a surprisingly large area, and you probably need to narrow down your base options and work from there. For me, at that time of year, you’ll need to base in a city, so York, or Durham make more sense. The weather could be fair or foul or just ‘meh’ in October (I’ve recently committed to a week on a long distance trail in Yorkshire at the same time, with some trepidation about what the weather will be doing) and the nights will be drawing in - I.e. it will be getting dark around 6.30 pm and won’t be light very early.
Some of your options may be limited too at that time of year. For instance the AD122 bus service along Hadrians Wall stops running on September 29th and some places will be closing for the winter - I’m thinking of National Trust properties like the Treasurers House in York and nearby Beningborough Hall, although I think they may be open until October 31st, you’d need to check. (EDIT: I checked - most are open until late October and beyond, but do still check they are open when you want to go - some closures any way).
I don’t think you can ‘do it all’ in the time you have available, not without it whizzing past the windows of a hire car, which is no way to see it! However, if you can firm up where you think you might stay, I’d be delighted to tell you what some of your options might be!
Hope you can get it sorted!
Ian