Two puzzles that occupied my insomniac mind

Two things kept me occupied whilst not sleeping this morning:

Let's turn off the stars for a bit. We're sitting in our spaceship and we look at the universe through a prism. Straight ahead and behind things are 2.7K. We turn on the impulse drive: we're now going 0.5c. Forwards is blue, behind is red.

We apply full retrothrust and by dead reckoning, we are stationary again. The Sky is uniform again.

Is the cosmic microwave background a universal reference frame (if not rigid)?

Then I wondered what it would be like to stand inside a perfectly reflective sphere of radius say 1 light second. (A BIG bobble :) If we were in the middle we would get very cold for 2 seconds, losing over a kW of heat. Then our heat would return. if we stayed in the middle we'd probably cook, but if we moved slightly, we would start to cool by photons going for long journeys.

What survival strategies would work in such a place? What is the steady state distribution after a spaceman has walked around the inside for a while and there are photons going every which way? What is the specific heat of vacuum? (I think the answer is 8π^5(kT)^4 / 15 (hc)^3)