2 ÷ 0.5 = 4 mg/mL. Concentration is vial content divided by diluent volume, so 2 mg of peptide in 0.5 mL of sterile water for injection gives 4 mg/mL. On a U-100 barrel one unit is 0.01 mL, so one unit of this solution carries 0.04 mg — 40 µg. That is the number to write on the label, because you will not reconstruct it from memory at an awkward moment.
Concretely, the distinction that resolves most of these questions is that bacteriostatic water suppresses microbial growth and does not sterilise anything. It buys you a multiple-withdrawal presentation; it does not make an unsterile preparation sterile, and it does not substitute for technique.
The order of operations matters: temperature first, then diluent measurement, then injection, then gentle dissolution, then labeling.
On re-freezing something that thawed in transit: if it arrived as a lyophilised solid that warmed but never got wet, re-freezing costs you nothing except the thermal cycle.
None of this is exotic. It is just the difference between doing it deliberately and doing it approximately.