We know that cosmic microwave background temperature is about 2.7K. But what temperature we will measure in space using a simple Kelvin thermometer in the shadow? Can it be lower than 2.7K?
Suppose a space ship is flying in our solar system with a speed of 0.99c relative to earth. Will it measure a different temperature? Much higher? And if so, will the intense heat cause the ship itself to warm up?
Re question 1: It's relatively simple to calculate temperatures in space because since there is no air, there is no heat conduction or convection. A body in space can only absorb heat by absorbing radiation, and can only lose heat by radiating.
The CMB behaves like a black body with a temperature 2.725K, so any body in equilibrium with it will also be at this temperature. If there are no other sources of heat your thermometer will have the same temperature as the CMB. However it's hard to achieve this. You could argue that you could shade your thermometer from the Sun, but whatever you use as a sunshade will eventually heat up and start radiating, and it will then heat your thermometer. The only way you'd get your thermometer down to 2.725K would be to put it in interstellar space or possibly even intergalactic space.
Re question 2: Travelling fast will indeed blue shift the CMB and raise it's temperature, and that will heat your ship. In principle if you travel fast enough the blue shifted CMB would vaporise your spaceship.
This actually happens to the Earth (well, not the vaporising bit!). The CMB is hotter in the direction of travel of the Earth. See http://apod.nasa.gov/apod/ap010128.html for details.