Black holes emit Hawking radiation regardless of ambient temperature. It's just that right now, the mass(-energy) of photons falling into the hole from the CMBR is outpacing the mass loss to Hawking radiation for large black holes, so large black holes aren't currently evaporating from their own Hawking radiation.

It's like standing next to a campfire. Your body is still emitting thermal radiation in the IR, it's just that you're receiving more thermal radiation from the fire than you're putting out.

No, the formula for the power of a black hole's Hawking Radiation is very straightforward, and it only depends on the black hole's mass.

I think the idea is that black holes can never shrink to a size small enough to emit a hawking radiation hotter than CMB as long as it receives photons from the CMB that that will balance the mass/energy of the black hole and keep it in equilibrium.

This obviously is not a problem if you create a new very small black hole from scratch as it only applies to pre-existing black holes that are massive enough so that their hawking radiation temperature is lower than the CMB

That's nice that you believe that, but your belief is wrong (for sufficiently small black holes)

Hawking radiation depends on mass of the BH, the smaller the hole the hotter it is.