1.Yes, not only that, also her school girl outfit materializes out of nowhere as well.
2.I don't know why it isn't possible that the people of Argo City couldn't follow the exploits of Superman including Clark Kent. It's not like anybody from an advanced planet couldn't have figured out Superman was Clark Kent.
3.It's possible she could have known about it from the events in Superman II. Very likely it would have been reported that Zod and the others escaped from the Phantom Zone. Of course, knowing about the Phantom Zone, and knowing how to access it are two completely different things.
In defense of the Supergirl movie over this though, it's fairly common even in serious movies for things that are virtually humanly impossible to occur: not only is it frequent that people are experts in doing things just from hearing from their existence (not the most serious movies, but, for instance, there was no way Lex Luthor could have reasonably known that kryptonite would be deadly to Superman in the first Superman, and in Superman III, it would have been impossible to put together a supercomputer like that in just a few days, yet alone construct it from simple designs written on paper towels.)
if movies had to be absolutely like real life, they'd generally be a lot less interesting.
I can't stand when people are confused about Lex knowing what kryptonite can do to Superman in the movie. Its been widely known in the comics since the 30s, that kryptonite can harm Superman, and Lex was a big fan of the comics. It makes perfect sense he would know. Lex even had a whole library in his underground lair. I'm sure there was a vast comic book collection included
And the truckers who confront her in an alley see her use heat vision on the knife making them drop it in pain but still decide to continue coming after her.
Basic intelligence needed to pass a driving test says that they should have retreated back to the truck and driven off at that point. They were lucky she did not kill them.