Why was Tommy allowed to be a psychopath?

Ultimately, Tommy doesn't get away with being a psychopath - he's killed at the behest of his superiors.

The Murders

For a time, Tommy gets away with murdering Billy Batts because his superiors don't know for sure that Tommy killed Billy Batts. If I remember correctly, only Jimmy (Irish) & Henry (half-Irish) are present for the murder. They're not made men (and cannot become made men because of their ethnicities); they're more loyal to their friend and co-worker, Tommy. Additionally, having helped dispose of Batts' body, they're also now complicit in his murder and could be punished, too.

Spider, on the other hand, isn't a made man, well-connected or a good earner, so his murder is of little consequence. Unlike murdering a made man, this (apparently) doesn't breaking any mob rule. Insightfully, Henry describes the mob power structure as follows:

All they got from Paulie was protection from other guys looking to rip them off. That's what it's all about. That's what the FBI can never understand - that what Paulie and the organization offer is protection for the kinds of guys who can't go to the cops. They're like the police department for wiseguys.

Apart from a few rules, like not murdering made men without approval and always paying the necessary tributes, wiseguys are largely autonomous. Tommy may be a psychopath, and he may not have many friends, but he's a good earner. Yes, people - as we see in the "you think I'm funny?" scene - are afraid of him. But Tommy hasn't committed serious offenses (that Paulie/superiors can prove) in the eyes of mob. Paulie gets his cut. Henry & Jimmy need him as a part of their cohort. So everyone is willing to put up with his crazy behavior.

Tommy & Spider

I disagree that there's an arc to Tommy's character. Apart from brief scenes of him as a teenager, I don't recall him ever being anything but a feared psychopath. Both Spider & Tommy are errand boys - but Henry, too, began as an errand boy. It seems to be a mob rite of passage.

I think the more interesting parellel is that Tommy, growing up, was bullied by Billy Batts (also, this is the only thing we get that's close to an explanation of Tommy's behavior). Now an adult, Tommy harasses Spider. This could be interpreted as ritualistic hazing or, more likely, as Tommy getting "revenge" on Billy via Spider.

That character was based on real life wiseguy Thomas Desimone. You can read about him at this link:

http://en.wikipedia.org/wiki/Thomas_DeSimone#References

Apparently he was as unpredictable, probably more than Joe Pesci's character. So it isn't so much that he's allowed to be psychotic. You could argue that being a mafioso is psychotic in itself. Why would his circle regulate that behavior when they benefit from it?

As mentioned in other answers, Tommy is based on a real person.

The mafiosos knew Tommy was an unpredictable psychopath, and feared that side in him (see the famous "Funny how?" scene). But they tolerated that side of him because he was extremely useful to them as a brute and as a hitman. No job seemed to be too violent or too dark for Tommy, and that made him a valuable asset to his bosses.

So when Tommy kills Spider (and does other objectionable things), the bosses are willing to overlook Tommy's violent side. After all, Spider was a nobody while Tommy was a valuable member of the mob.

However, when Tommy killed Billy Batts, he crossed the line. Killing a made man without getting authorization first is one of the biggest taboos in the mafioso circle. It disrupts the whole social structure of "being made untouchable". So that was a transgression that his bosses could not ignore, no matter how useful Tommy was, and the order was given to whack him.

Everyone knew he was psychotic (and I mean more so than others although the mafia is full of dangerous people) - note how someone (not sure who) tries to intervene in the 'you think I'm funny' scene - he says something like "Tommy, you got it all wrong" and Tommy replies "He can speak for himself" and everyone is immediately quiet as they did not want to become collateral damage, even his one of best friends is not sure at first whether Tommy is about to turn on him. However, it is revealed to be a big joke and everyone laughs with relief and because they want to stay onside with Tommy. Seriously, would you have stayed dead pan and reprimanded Tommy?!

Murdering Spider did not really matter to the mafia, he was just a low level errand boy. He did not have any connections etc so the mafia are unlikely to care whether Tommy killed him or not. Remember these people would order an execution in the same way I would delete some code and retype it.

At that point Tommy was useful, he was charasmatic, brave, resourceful, terrifying but apparently controllable. He seemed to largely do what he was told but then 'they' found out he had killed a made man and that meant a death sentence.