Can I call an assistant professor "Professor"?

Yes, you can call any kind of professor a professor, and you should. Addressing someone as "assistant professor" or "associate professor" would be...well, it's simply not done, so I can't say if it would be rude or just weird.

It is similar to military protocol, actually: e.g. if someone is a rear or vice admiral, you call them admiral. In contrast to the above example, I'm sure some knowledgeable party could be specific about what bad thing could happen to you if you screw that up!

Yes you can.

The "Assistant Professor" still teaches. The word Assistant is there to denote the rank within the academic system. Some have taught longer and are more accomplished and are rewarded accordingly.

Actually, calling the person "Assistant Professor Jones" would be very awkward and cumbersome. It should be avoided.

Yes, you can call an assistant or associate professor "Professor." That's completely normal practice. The only time it might be inappropriate is if you are writing them in a formal context. "Prof. Smith" is always OK, but saying "Mary Smith, Professor of Unusual Studies" isn't really appropriate if Smith is an assistant or associate professor.