Refused entry to UK, not sure what to do

A removal from port is one of the more distressing experiences a traveller can have, and the UK Border Force has been taking a harder line towards abuse. Based upon what you wrote, you were building up a private life in the UK and working without the proper documentation (even though your work was performed for US companies and probably denominated in US dollars). Your narrative also states that you were detained rather than given temporary admission; this would indicate that in addition to failing the landing interview they concluded that you were likely to abscond. That usually means they have palpable evidence that you will continue to abuse the system and try to go underground.

What to do now? They gave you some forms as part of the removal process. Hold on to them because you'll need the reference numbers in the future, essentially for at least 12 years or until you acquire citizenship in an EEA member state. The forms will also say if there are any bans in effect or if you have been classed as undesirable. This is unlikely because your narrative is a fairly common one, but you should check nonetheless. They also took your biometrics and they will keep those on the system for 12 years. They will send the information to the EEA member states, the Commonwealth members, and to the DHS (regardless of your nationality). The other step to take immediately is to get in touch with HMRC and find out if you need to file a return with them.

Is it possible for me to ever get back to the UK? Yes, you can try any time you want at any port you want (except through the Common Travel Area as you no longer have that option). The 'golden advice' in these situations, however, is that you get an entry clearance before attempting another entry. It's not required, but is the only way to assure that you do not endure the discomfort of another removal from port. Note: trying to use the Common Travel Area to access the UK is a dreadful mistake, don't do it.

Should I apply for a visa from the British Embassy? Yes. As mentioned, it is not required, but a highly recommended option. You will need to enrol your biometrics again and submit your passport along with all the necessary evidence to qualify as a visitor.

Will this really take three weeks? It's possible. On the other hand if your evidence is weak and you plainly do not qualify, it could take as short as a day to get a refusal back from them. Constitutionally, they can take up to 90 days to reach a decision, but based upon everything you wrote, your case is relatively straight-forward. It's straight-forward as long as your paperwork with HMRC is in order and your bank statements do not show deposits from working during the time you were in the UK; if they do, then it becomes complex very quickly. It goes without saying that you should do everything in your power to assure your application is successful.

Should you cancel the Doc/Fest? There's no need to decide that today, but there's definitely a nascent possibility that 30 days (when the festival starts) will not be long enough to get everything sorted out. If you didn't have the funds at port, how will you get them in Paris?

For the last part of your narrative, when the Immigration Officer told you "We make the decisions", he was right; but I think it was a communication problem and he thought you wanted to make an in-country application. With or without an entry clearance, you will have to clear a landing interview every time you arrive, and the IO has the final word. Sidenote: trying to get consular assistance from the US Embassy (or any other diplomatic mission in the UK) in a removal situation is beyond worthless since they can't do anything and might charge you for it later anyway.

Finally, lots of people find it useful to instruct a member of the Law Society to represent their next application. But you will need to clear their money laundering hurdles (which will be difficult if you have been working illegally) and pay their fee up-front (I'm guessing £800+ based upon what you wrote, but that's strictly a guess!)

The bit about "meeting with Artistic Directors of various theatre companies to see if they might be interested in producing my work" is in the work spectrum.

If you were trying to enter the UK in the visa you obtained at the airport and valid for 6 months, and claimed that you were a tourist, then this could be the reason. Both assertions are not compatible. Add to it the fact that you didn't have enough money (from their point of view) for the two months stay.

The Eurostar route won't work for you. Obviously, you won't be send back by plane if you try to enter by train, but that's the only difference.

Since you were denied entry, you'll need to apply for a visa, even if citizens of your country are normally allowed visa-free entry.