Is it legal to alter the login screen of the client?

I am not sure about this, but my guess would be that you can't be banned by doing so, since it doesn't gives you some kind of advantage and is purly for cosmetic reason. People are using custom skins for years now and riot stated that you won't be bannes for using them, so I don't see a reason why it should be different for a custom login screen.

Edit: But maybe post this on the “Help & Support“ board if you are still unsure, you might get your answers there.

No it shouldn't be. Since you don't harm anybody by doing so (eg. using third party programms such as scripts).