10 ÷ 2.5 = 4 mg/mL. Concentration is vial content divided by diluent volume, so 10 mg of peptide in 2.5 mL of 0.9% sodium chloride 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.
The answer depends on what you want your measurement resolution to be, and that is a real trade-off rather than a preference. More diluent gives you more syringe marks per dose and therefore less rounding error; it also gives you a larger volume to keep cold and a longer period over which the solution has to remain within specification.
Room temperature before drawing is worth the ten minutes. Cold solution is more viscous, draws slower, and is more likely to pull a bubble past the plunger seal.
Tilting the vial to pool solution in the corner before the final draw, and giving it a minute to drain down the walls, genuinely recovers ten to twenty microlitres.
Worth saying: if your arithmetic and someone else's disagree by a factor of ten, one of you has made a unit error.
None of this is exotic. It is just the difference between doing it deliberately and doing it approximately.