Had Brake fluid flushed.
had brake fluid flushed, new pads, brakes work after 1-2 min. of car warming up/running, hand brake does work to stop car

sounds like you could have some water in your brake lines that freezes l would start by bleeding the brakes than expecting the brake caliper to see if there is ice or and obvious defects on it

YOU NEED CHANGED THE BRAKE FLUID FOR NEW ONE.
BECAUSE THE OLD ONE WAS CONTAMINATED AND HAS WATER IN THE BRAKE SYSTEM.

It is a long shot that ur brake fluid has become contaminated enough to cause this problem.
Brake fluid is hydroscopic & water, if any will mix with it. You would need a huge amount of water in the brake lines for this to happen.

l have operated cars in temperatures as low as -30F & have never had a problem with my Civic.

You have brakes, it may just feel very stiff due to the change in temperature. Your handbrake will still work, perform a small test in ur driveway?

If water freezes at 32F, then -10F is really ice cold temperatures. If ur brake pads or brake rotors r exposed to any moisture, water, or snow, that immediately can become ice at such temperatures.

If ur brake pads r no longer rubbing against metal or ceramic & instead ice, it may explain why u have no stopping power in such weather conditions. Ice is slippery & allows little friction.

However, there r high performance brake pads that ''might'' work for u & r supposedly functional in extreme cold weather. You can try switching to those brands. Not sure how they work but it is possible that they fuse salt into the ceramic padding which may help reduce the freezing point of water.

Good luck!