Probably cheaper to fly *home* from Paris; there are extra taxes when flying from the UK now.
Not sure how cheap you're trying to do, but check out Youth Hostels - you can book a whole room and that might sleep 6.
2 weeks is not a long time to do 'Europe'. You might want to, say, just do the UK, or just do France, or whatever. I know the distances are shorter, but there is just so much more stuff packed in (ie, London to Birmingham is what, 2 hours? Birmingham being the UK's second city...).
You can get special rail passes *in advance* that will allow you to cover a lot of distance. If you're going to rent a car, bear in mind everything will be manual transmission.
There are many many cheap airlines operating all over - EasyJet are (IMNSHO) much better than RyanAir, but either way they are *cheap* - if you work around when their cheap flights are.
Ireland is awesome - 2 weeks you could do a nice loop from London, through Wales, over to Ireland on the ferry, up to Northern Ireland, over to Scotland, and back down to London. Even then you'd be travelling quite a bit.