Generally living organisms become bipedal when they have another use for their legs other than walking. See birds. In our case it was the usage of tools.
My interpretation is that the second pair of legs (arms) only gets in the way, if you want high speed on a plain. That is true for both theropods and primates, just look at a chimpanzee running...
Horses and cheetahs both seem to do fine, and much better than humans.
Also, the human body-plan seems set up such that a running human has dynamic instability: we run "better" by pumping our arms around, keeping balance with the shifting mass rather than purely relying on our legs to do that job.
Horses (and various ruminants) are a bad example because they need a relatively big mass of fiber in their digestive tracts. Can't do that on two legs. Still, their fore legs are considerably lighter and contribute relatively little to forward motion.
For that matter, a human can run down a horse. Some Africans still do that with zebras. Cats can't do that.
And no, "pumping" arms around is not really a necessity.
Again, Humans have done this as a hunting strategy for at least as long as there is H. sapiens sapiens. Pick an individual animal, run after it until it collapses.
I guess the horses they hunted weren't as well trained, bred and fed as those in sports competitions. Wild horses in general evolved to run occasionally over relatively short distances.
They run better over short distances. Bipedal great apes do better over long distances. This is proven both by experienced hunters and myself as I hunt down my cat for kisses. After just a few minutes, she hunker down and accepts her fate since I can literally jog around my house all day.
Is there an evolutionary story for why T. Rex ended up bipedal with mostly-vestigial forelimbs? It works fine for their ecological niche, but I can't picture the evolutionary intermediates between them and a quadrupedal ancestor.
Actually, there's a very simple reason: the T. Rex had stubby arms so it could have a big head.
The T. Rex started out bipedal, with functional arms -- you can pick that as its starting point, there are plenty of those guys running around in the fossil record. As its head got bigger and bigger, it became top-heavy: its back end had to balance its front end when running, or else it would just face plant constantly.
If I had to guess (I haven't actually looked into how much of this we have recovered from the fossil record; I just happen to have heard the reason above) the T. Rex's head probably got pretty big first -- for a while, bigger heads could be counterbalanced by bigger tails while still retaining normal-sized arms. But there's a limit to how big animals can get, so eventually the proto-T. Rex would have the biggest tail it could support.
By this point it would be firmly in the big head niche: its diet, survival strategy, and sexual selection would revolve around using its giant head and powerful mouth to just bite the crap out of everything and generally be awesome. Now come the tiny arms: even when its arms were still functional, they were less awesome than its giant head. Trading slightly smaller, slightly less functional arms for a slightly gianter, slightly more awesome head was a win at every point down the curve, until it was stuck with tiny, useless arms and a really awesome head.
The current thinking is that T. rex used its stubby arms to grasp prey when it went in for the kill. A similar line of thinking applies to the flapping wings of birds: earlier theropods flapped their forelimbs in order to help hoist themselves on top of their prey to make the killing bite.
Their common ancestors may have evolved bipedalism because forelimbs proved cumbersome when taking down large prey.
We're dealing with larger than mountain lion scale beasties here. Fun fact about hippos: they can't jump. Like at all, unless they're submerged and can use the buoyancy of the water. The same is true of just about any modern tetrapod within that size range. If you're a tetrapod the size of a mountain lion or even a bengal tiger, you can still have speed, agility, and the ability to spring, pounce, and grapple with the forepaws. At T. rex scales, the square-cube law has something to say about that.