re: "Can I have several contracts with one client at the same time?"
Upwork places no restrictions on the number of simultaneous contracts you can have with the same client.
re: "Can I work on a new project under another contract? Or should I continue under an existing contract?"
That is really up to you and your client.
Personally, as a freelancer, I steer my clients toward closing contracts and creating new contracts when we have different projects, or even different phases of a project.
a) the contract titles are more accurate
b) the clients' bookkeeping is easier, with my billing being attached to the right project
c) I get additional feedback and job counts on my profile
(Although I don't cite C as a reason when discussing this with clients.)
Yes. You can.
But, if one contract "goes south" with the client and they close the other contracts, this could slam your JSS (just a heads up).
If you're on an hourly contract, just have them continue with that single contract (recommended based on the possibility of the scenario above). If it's fixed rate, then they can add additional milestones (I mean, unless you've cloned yourself, you're only able to work on one project at a time...meaning, you'll work on Project A for a while, and then work on Project B -- but, it won't be at the exact same time).
I believe there is an option under hourly for the client to set up different project categories (Petra? Preston? Can you confirm?).
Of course, it's entirely up to you and the client.
Mykola, if you are working on a fixed price contract for the client and the 2nd/3rd/4th, etc. contract involves a different perspective or role - again, yes you can have different contracts with the same client.