Same way Tony Stark got Peter Parkers home address.

The magic of bad screenwriting!!
Taaaaaaaadaaaaaaaaa

They were a match on tinder at the museum
How can Superman fly, while carrying a kryptonite spear?
It's happened before,
the end of Superman Returns has him flying while lifting Lex's entire "new country"
of course i'm damaged, it's tattooed on my forehead isn't it?

That's a great point lol. Obviously superheroes like Batman have a secret identity, but there's no reason to assume Superman would.
http://www.astortheatre.net.au/
Support great cinemas.

Oh, I was speaking from the man on the street's and Bruce Wayne's perspective. Who says Superman puts on a fake beard and punches in at work somewhere?
Also, this
:
http://www.imdb.com/board/bd0000130/nest/263626706

If Superman doesn't use a secret identity, then the Daily Planet has a hell of a diversity-in-hiring program.
Seventy-seven courses and a regicide, never a wedding like it!

Who said the man who wandered the Earth as a drifter until finding a spaceship (as Lois' leaked story tells) and now flying around in a red cape now uses a secret identity?
Also, this
:
http://www.imdb.com/board/bd0000130/nest/263626706

But he can't figure out Superman's secret identity?
If you can't defend a movie without bashing another, you will be put on ignore. No exceptions.

Batman can get anyone's email address.

He's Batman. He didn't run into her again at the museum by accident.
Also, this
:
http://www.imdb.com/board/bd0000130/nest/263626706

Clearly Diana Prince was someone of note. She was at multiple high society functions and the airplane attendant knew her by name. You don't have to know she is WW of Themyscira to know she is a socialite or historian or what have you named Diana Prince. As such, it wouldn't be hard for Bruce to get her email address.
As for flying with kryptonite, each version of Superman can be flexible with its own mythology. There is no one correct way that kryptonite affects Superman. There are versions where it hurts him greatly but it doesn't instantly remove his powers. After all, he has survived against the likes of Metallo, Kryptonite Man, and Lex in his kryptonite enhance war suit. Even after inhaling the K gas, which instantly affected him, he still took many blows from Batman that anyone of them would have instantly killed a normal person. Further evidence he doesn't lose all of his powers instantly.