Yes. One place in space has different temperatures. I would assume even individual particles are not distributed by a Maxwell distribution, so the concept of temperature is hard to apply. The background radiation has one temperature. If you add the sun, however, you already have a problem as the sun radiation is not in thermal equilibrium. So depending on how you look at it, you get different temperatures. The particles have a high energy, so also a high temperature. But they are so rare, that radiation is the dominant mode of heat transfer and determines the temperature of a thermometer placed in space.
No, to orbit the earth at an height of let’s say 1000 km you would need a speed of around 7km/s. If you go faster, you don’t follow an circular orbit. Wirh around 11km/s you would be so fast to leave the gravity well of earth. The particles in those colliders are almost moving at the speed of light. To be exact, they move only 3.1m/s slower than the speed of light, so almost 300000km/s. They would fly almost straight and would be barely influenced by the gravity well.