In terms of names, it will only be known as Bramley Moore if it's played in for a couple of seasons as that name first to establish it.
Otherwise, if it opens with a sponsor name it will always be known as that, even if it changes.
For example, who still calls it the Echo Arena even though it's been the M&S Bank arena for years now?
Who still calls it the Reebok Stadium even though it's had two names since.
The next sponsor to Arsenal's ground will have the fight against literally everyone calling it the Emirates. (It was called Ashburton Grove whilst being built)
Likewise the Alliance Arena.
Spurs have been clever calling it the Tottenham Hotspur Stadium. They intend to get a sponsor eventually but wanted to establish it first.
We need the money though it remains to be seen if it goes by the name of Everton Stadium.