They sure would, as long as 'uncycled' doesn't mean 'lots of ammonia'. =B
Shrimp don't tolerate ammonia/nitrite spikes very well, but I stuck half a dozen in my 3.5 gallon before it was cycled and the tank cycled with them in -- all were fine (except the ones my betta girlie ate..). I was pretty scrupulous about keeping the ammonia down, though, and the tank is way heavily planted, which helps shrimp in several ways. (if you look at the album for my dymax 3.5 you'll see what I mean)
They do require a filter and a little bit of water movement - mine do fine on sluggish betta-friendly flow.
If you have a filter, then your tank WILL cycle. The problem with small tanks is that the cycle is very hard to keep stable. I find this is a great deal easier if the tank is stuffed with plants, primarily mosses and other ammonia sinks.
You can expect your betta to eat a few shrimp. Some are better at hiding than others, and they're much easier to catch, apparently, when larger and thus more visible, I'd assume.
Lots of java moss/hidey places is highly recommended. And when they run out of wood gunk to eat, they might need some shrimp pellets, though mine prefer to eat algae and barely touch the pellet food. :)