Toughy for that price. You might be best to take a look at (for example) some refurb models. You can get Dell Inspirons for £275 with a Core i3, 8GB RAM and 1TB HDD if you don't mind the odd cosmetic mark. Anything with a Core i5 or SSD is likely to be out of reach price wise unless you go for a second hand one.
Such laptops should be capable of a few basic games (especially slightly older ones) if you're careful with the graphics settings.
Toughy actually. Until Ryzen, AMD have been unable to compete at the high end, but their budget CPUs weren't bad. Looking around, performance of that CPU is almost identical to the Core i3-7100U in the Inspiron I linked to earlier, however with the Radeon graphics bolted on, you'll probably see better gaming performance from it.
Given that it's £25 cheaper as well, the AMD one might be worth considering.
Yep! Now with Ryzen around, it's even more difficult as AMD and Intel trade blows at the higher end too with both sides better at some things and a bit worse at others. AMD currently scores bonus points due to being a shitload cheaper as well. But for none-Ryzen chips, yeah! It depends...
Ooh I hate half-sized shift keys like that. Had one on an old laptop once and every time I tried to type a capital letter, I moved my cursor to the line above instead :(
My Lenovo keyboard is decent enough but it has the Ctrl and Fn keys the wrong way round. Luckily you can swap them around in the BIOS. It's fun to tell people to press Ctrl + something and them wonder why it's not working.