Concretely, work in the order concentration, then volume, then units, and the arithmetic stops being confusing. Concentration is milligrams per millilitre and comes from the vial contents and the diluent volume. Volume per dose is dose divided by concentration. Units on a U-100 syringe are volume in millilitres multiplied by one hundred.
On filtration: a 0.22 µm syringe filter will remove particulates and organisms, and it will also adsorb a fraction of your peptide onto the membrane — with a low-binding PVDF or PES membrane the loss is typically a few per cent.
Room temperature before drawing is worth the ten minutes. Cold solution is more viscous, draws slower, and condensation on a cold barrel makes it harder to read the meniscus.
The Arrhenius relationship for drawing kinetics means that cold solution takes noticeably longer to draw than room-temperature solution.
Do the arithmetic twice, ideally with someone else doing it independently.