changes in output to the oil rigs / refineries themselves can easily be ramped/up and down, the problem is the industry relies on a complex network of "smaller" offshore construction companies[1] which are not so flexible in the sense that they can just stop without repercussions. They supply a range of services such as inspection, pipe-laying, testing and connecting actual well-heads[2] using ROV's and deep-sea saturation divers. Staffing these jobs is highly volatile since most people working for them are also freelancers flown in from all over the world. When they bid for these projects (with Shell, Conoco, and others) it takes weeks/months to mobilize their personal, ships and technology for the job.

Usually when there is a reason to reduce capacity on a rig (mostly due to weather) it's normal to just stand-by (sometimes for weeks) until they can continue with the job. When I worked offshore it was never possible to say when I get back home, e.g. the job was estimated to last 10 days but could end in 4 months. And once we were done we were almost always flown to a new destination (so they would try to keep us out there as long as possible). While places like the North Sea regulations would be strict (28 days offshore would be max before you have to go back to shore), in the Golf of Mexico, or in Asia we would never have such rules. I know people who spent months locked in the compression chambers (sat system) or until the project was over. And none of us dared to say we want to go home (it would be career ending).

[1] Oceaneering, McDermott, ...

[2] https://en.wikipedia.org/wiki/Wellhead

What do you mean by "require freshly pumped oil"? The contract is for a commodity at a certain price at some time in the future. There might some quality requirements but other than that nobody cares whether it comes from storage or is freshly pumped. It is beneficial for the buyer that the seller has ample storage because it means less risk that the seller defaults on the contract.

I mean I could imagine some contract where party A agreed to continue to accept/transport/"deal with" all the oil party B produces, and is required to continue even if they are losing money on the deal.

I'm not sure why anyone would have signed such a contract, which is why I called it weird, but it does seem plausible.

If you buy a futures contract, you are required to accept delivery of some fixed amount. You don't have to take "all the oil party B produces", just what you agreed to ahead of time. This protects you from upward price trends, and the producer from downward price trends.

Sure, but if the cost of short term futures contracts is negative or zero the oil companies would just buy them back instead of continuing to produce. When we say the cost of oil, we really mean the cost of short term oil futures.

Government productions contracts might require fresh production. No one else would realistically insist on that.