Wow, nice dance number there, but if it took place in a massive auditorium behind the club's stage, and there was no way to televise it back in 1935, who in the club was even able to watch the performance?
It might be Willie's Imagine Spot. This is actually suggested in James Kahn's novelization.
Or just a joke (at the fourth wall's expense) about how they used to shoot extra material (or use recycled material from earlier films) and splice it into movies to make them longer. The obvious break and different quality in video would play into that sort of meta-humor. Remember that these films are send-ups of old pulp serials.
Or a more mundane explanation, there's a movie screen in there and they basically cut to a much bigger pre-filmed dance number that was being projected on the screen.
In colour in 1935?
Kinemacolor was invented in 1908 and Technicolor was invented in 1916. So yeah, maybe.
That Willie interacts with?
Lao-Che, Lao-Che... after you brilliantly got Indy to drink poison, why tell him? If you'd just kept him talking for a few minutes you could have simply taken the diamond off his corpse.
He would have, but he caught the Villain Ball offscreen and decided that simply letting the poison do its thing wouldn't be diabolical enough.
Also, Indy embarrassed the family by injuring and nearly killing Kao Kan the previous night, which probably left them furious. Lao doesn't just want keep the diamond, he wants to pay back Indy by beating him, making sure he knows he beat him, and gloating about it to his face. Doesn't end well for him, of course.
As annoying as Willie's constant shrieking can get, remember that she's just an average woman who until recently was living a very pampered life caught up in what at many times is an utterly terrifying situation, and suddenly her reactions are a lot more understandable.
Temple of Doom is also a prequel to Raiders Of The Lost Ark, which means that Indy being confronted by TWO swordsmen takes place before the famous incident in the marketplace. Indy's gun could've come in handy several points through Temple, and thus it's his first choice in Raiders when a sword-bearing enemy comes across him.
It's also why he appears to carry two guns in Raiders.
Another is ripping out a beating heart. It's all about perspective. When we see it done the first time, it's from the followers' viewpoint, who believe the victim's heart is being ripped out. When we see it tried again, it's from Willie and Indy's perspective, who realise the whole thing is a sham, and don't fall for the trick.
Not to mention the fact that your argument is built on the assumption that any supernatural element in the story means that anything which could be supernatural is. Point in fact, if the heart-ripping really is the result of the power of suggestion(as I am inclined to believe), it makes the actual power of the Stones more spectacular and goes further to explain why the Thugees would want them, i.e. so they no longer have to pretend to have magic powers.
Plus, all the followers were drugged with hallucinogenics so they would believe whatever Mola Ram wanted them to believe.
There's also the factor that he probably didn't rip Willie's heart out, because he wanted to do something more terrible to her. Such as making the brainwashed Indy restrain her and lower her to the pit.
Why the stones worked in Indy's favor even though he didn't worship Shiva. The Shaman prayed to Shiva prior to their departure to Pankot to help guide them. So Shiva was on his side, whether he worshiped Shiva or not. As well as the fact that because Indy spoke Hindi, he knew how to invoke Shiva through the stones just by knowing the language. Whilst Mola Ram who was aware of both Shiva and Kali pretty much tarnished the former. Gods get more pissed off at believers tarnishing their truths than those that don't believe in them at all. In addition, Shiva specifically has been known to acknowledge worship even when it's inadvertent and the worshiper doesn't know or realize what he's doing: there is a Hindu story of a thief who receives Shiva's blessing after the thief spends the night hiding in a tree. The tree is a kind sacred to Shiva, and the thief spends the night idly pulling leaves from the tree and dropping them, where they land on a Shivalingam together with milk dripping from the thief's sack of food. If Shiva will acknowledge that, Indy's heartfelt denunciation of Mola Ram in Shiva's name would likely qualify as well.
The Artistic License regarding Kali worship makes more sense when you consider that Mola Ram is blaspheming Hinduism in-universe.
That and the possibility that he doesn't truly believe in the god he claims to worship and that he's simply using his self proclaimed belief to control the other Thugees. Like a real cult.
Also: this is 1935. As multiple people state in the movie, the Thuggee have been extinct for a hundred years at this point, plenty of time for their image to get distorted. So, picture this - Mola Ram decides to revive the Thuggee as his own personal cult, he bases it on the information that's available to him at the time, but because so much of that information is British colonial pop culture and propaganda, or influenced by it, the cult ends up looking more like that than like the reality.
Why Willie didn't burst into flames when she was closer to the Lava than the last guy? Because at the time the other guy was being sacrificed, Mola Ram had his heart out and was chanting. It was a spell Mola Ram chanting which caused the flames to rise. Whereas when Indy and Short Round were trying to save Willie, Mola Ram escaped and the rest of his chanting followers were busy fighting Indy to invoke Kali.
The Thuggee assassin tries to strangle Indy in his bedroom. This is a reference to the way Thuggees were said to have killed their victims: strangulation.
Also for a rather hefty man, the assassin seems to blend in with a flat, two-dimensional wall painting a little too easily...
Of course, Indy was too focused thinking about Willie.
This movie chronologically takes place before Raiders of the Lost Ark. Where happened to Short Round, Indy's KID sidekick, in between the two adventures?
Word of God is that he went to boarding school and became an archaeologist.
Which also nicely explains why Indy does indeed care about the money he gets selling the few artifacts he can to the museum after Belloq steals the idol in Raiders; He has Short Round's schooling to pay for.
You have to wonder how many of the other Thugee cultists were just innocent guys being mind controlled like Indy was.
Doubly so when you consider that the shooting script and the novelization showed Mola Ram himself snapping out of brainwashing just before falling to his death.
You also wonder if the sacrificial victim was a cultist randomly chosen, or one of the enslaved children (he almost looks like a teenager).
When you consider that being burned breaks the Black Sleep, think of the poor cultists who ended up in the lava pit. They presumably were broken out of their mind control just in time to fall into lava. It would be more horrifying if the Thuggee slave driver also broke free of the black sleep once he started to get flattened by the rock crusher.
Mola Ram cavalierly kills off his own men regularly. It's not too surprising when you look at it from his perspective... these cultists mean nothing to him, and with the hundreds of captive male children (which will eventually become men), he has an inexhaustible supply of mooks. So their lives have absolutely no value to him.
There is a strong possibility that Mola Ram's men were once children who were kidnapped and enslaved like the recent ones. Then end up becoming nothing more than brainwashed minions by the time they've reached adulthood.
Just what exactly happened to the prince prior to being brainwashed?