Take this all with a grain of salt since I wasn't in the beta so all I know is what I've learned in the last day, but:

re: seeing the cargo in the Ai tab
This only applies when the ship actually has a load and is on it's way back to the destination planet, it'll show how much cargo it's carrying. Otherwise, it'll show Harvesting or something.

1) Correct
2) The collection rate is based on the ship's cargo space. The harvesters you can build starting out have 250 space, so whether the planet has 10k or 100k, it'll make no difference in the collection rate.
3) Assuming it has a mineral production rate (10Ku/+1Ku for instance gains 1Ku per hour), it'll never run completely out, but once you run out of the stored minerals, you'll be getting less than a full load every trip as you'll only be picking up however much has been produced since the last load
4) No Idea