To join the war, FDR paid the Japanese to attack so they have reason to intervene. Is there any plausibility to this?
Edit: took out an unnecessary chunk
No not really. To the plausibility of it, yes they exist.
There were decades of building animosity and growing unease on both sides of the Pacific, and frankly for 30 years the main enemy the USN envisioned fighting was Japan. This long predates any one politician, and FDR knew very well about the Navy's state from his time as Secretary.
He was also committed to a Europe first policy, which the American publics fixation on, and early setbacks in the Pacific hampered by requiring the diversion of additional resources.
While there were elements of the govt that at least thought about poking the dragon to get the war going as evidenced by the McCollum Memo, frankly FDR's actions do not bear out his being "in on it" or a larger plan at works. He simply did feel strongly about the wrongness of Japan's actions, and wanted to project strength in both the embargo and military positioning.
We should also consider that if the plan was that Nazi Germany was the greater foe, as FDR and Churchill agreed, then starting the war by having the arm which had hold off Japan, the navy, start the war demoralized, gutted, and with their primary striking force(at the time still the battleships), all sunk, that is not how you start a war to win it.
Your main question is fine, but the part after "Also, ..." makes this look like a trivia question, which are not allowed in this sub. Could you edit that part out? Thanks!
There was a "conspiracy theory" at the time but it was not what you've stated. Basically it implied that FDR purposefully set up conditions that would require the Japanese to attack, and that he had forewarning of the attack but chose to ignore it because he was eager to have the United States drawn into World War II, and it would provide a good casus belli.
Wikipedia has an OK article on the subject. You will see it dates from the time of the war itself. I have never seen any theories that suggested that somehow FDR and the Japanese were in "cahoots" — there really is no need for such a suggestion (Japan's imperial ambitions were explicit to say the least).
Mainstream historians reject that Roosevelt had prior knowledge, but the notion that FDR deliberately maneuvered things in the Pacific to possibly provoke an attack is not an entirely implausible interpretation of US actions over there, though it does ignore the fact that it takes two to tango (if FDR held out "bait" by creating circumstances that put some constraints on Japanese expansion in the Pacific, the Japanese eagerly took it).