Does the income from this count as pay?
No, the government is totally happy for yo to not pay income tax. Actually when you grow and become a hugh advertising income business like google, you never pay any tax.
Heck, do you really think the government is stupid?
You have to pay income tax on YOUR INCOME. Not "on your income except", ON YOUR INCOME. Donation are nothing more than income, otherwise I would happily "donate" my work to my clients in return to them donating their moeny to me.
I agree with your obvious assessment that the governments are mostly run by idiots, but sadly I have to inform you this idiocy does NOT extend towards ignoring income that can be taxed ;)
Do I need to register myself as a trader, or create a company?
Different question and depends a lot more on what you like to grow to and other elements of your strategic setup. In the UK a LTD is cheap to set up, though, so I would just do it.