Is it illegal not to have a name?

No, but it's very inconvenient.

And if you tell people you don't have a name then you might still get arrested.

do you want a number instead?

No
Its a good deterrent against the deathnote after all

I have no idea... I have never really thought about it before.

Nope, because it´s like if you didn´t existed.

Yes

Kind of awkward but I don't think it's illegal.

I've never thought about that...but it probably is. If you think about it, don't you always have a name because how will people address you? Unless you don't live near people and aren't aware of yourself.