Joining the Axis early gives a research bonus, but you lose the bonuses in espionage, so that's not a clear "win", in my opinion, more like a fair tradeoff. The problem with joining a faction is that your Threat is magnified for members of opposing factions, meaning that they will see lower Neutrality due to Threat, reducing Consumer Demand, and leading to more IC available to build military units as a result. Other than the tech boost, I see joining the Axis early as a marginal idea at best, especially if the war with China drags out.
Raising Threat on the UK or France will bring Italy into the Axis ahead of schedule and significantly reduce Consumer Demand for all Axis (and Commintern) members, and to all non-democratic countries to a lesser degree, whether you're in the Axis or not. Since I've generally got spies in France either stealing technology or reducing Neutrality, I typically target the UK to raise Threat, and their naval and aircraft buildup amplifies that Threat.
Getting involved against the Soviets is a big question, which for me will depend on how things go in China/SE Asia/India. I can probably cut the main supply routes from Moscow to Vladivostok, and eliminate the Soviet divisions in the peninsula without too much trouble, but crossing all of that horrendous infrastructure to do much beyond that further to the east seems daunting....and then the Soviet tanks will likely show up in force.
Note about cancelling trade routes: the amount of dissent you get from having convoys sunk depends in large part on how many convoys you have running. The more convoys you have, the less an individual sinking affects your NU. Don't cancel TOO many convoy routes, only the ones most at risk and not very essential.
Incidentally, my Japan campaign (launched on July 2, 1937) is now into October, and so far I've taken out Shanxi, pushed south of it all the way from directly below ComChi to the coast (except for the Qingdao peninsula where the capital relocated to, and which was just cut off from the rest of the country). None of the warlords have joined yet. A second front was opened just below Shanghai, and the port seized, followed by a push inland which took the capital city of Nanjing, forcing the relocation to Qingdao. That southern enclave has now expanded to just about the limits of the manpower I was able to assign to it, and is now waiting either for 3 additional divisions to complete around the end of the month, or for the northern front to cross the 3 remaining provinces between them and link up. As long as I don't take Qingdao, it should make Shanghai the new supply depot, but I don't know for certain whether it will change it again once the new capital falls. The Chinese navy is either eliminated completely, or down to a single damaged sub. This feels less like a war, and more like an execution.