Yes you can extend 6m and keep the eaves at the party wall line at a max of 3m high, this falls under permitted development with the caveat that your local authority will liaise with all of your neighbours - if any of them are unhappy with it then it will be turned down and you would require full permission. My advice - go and see your neighbours with drawings and keep the height of the party wall down, 3m is high; 2.4m can often be more acceptable David

Your best bet would be to call them directly. I didn't think so but a quick google search reveals that you do need planning permission in some examples

Hi do I need planning permission to lower a garden by 2 feet high? Thanks

@christopherblowers any wall / part of a wall that straddles or touches a property boundary, external or internal, is a "party wall" and work on those beyond notional (that is, significantly more than putting dowels and nails in) must notify the neighbour. They cannot object but for extensive work, they can request you to pay for a survey (to record the condition, in case the build causes them damage). If they're OK with it then they can just consent to the notice and you're fine - keep both the notice and their written consent with the property documentation. As Tariq says for small things like this you can find the notice forms on the Internet and send them yourself.