I was born a couple of years after this film. I find with many films of the 40s and 50s, there are aspects of those films it takes me a little time to understand. The culture, what the audience was expected to understand without it being explained, the consensus of what was normal, it takes me a bit longer to figure out. But I love those films because of that.
I think Arkadin was so devastated when he heard his daughter tell him she had already spoken to the investigator, when he believed she now knew of his shady past, he was in shock. So he failed to land the plane and it ran dry on gas and crashed.
I am presuming that since the investigator made it to Spain, he was off the hook, he was out Germany and so out of the reach of German law enforcement. Presumably he was able to go his own way.
Before he told the girl the whole story, she didn't know he had tried to blackmail her father and she seemed to like him. Once she heard the whole story she soured on him, understandably. But once he learned Arkadin was out to kill him, he had to tell the daughter in order to take away Arkadin's reason to kill him (or so the story line goes-I would think Arkadin would have had a greater reason to kill him at that point).
These old films beat most of the modern films, I think. It is so easy to make a film these days, there is a lot of junk out there.

If you are looking for a fast paced, modern special effects horror film don't expect to find one her. If you are looking for great acting and a compelling story give it a try.