Short answer: The testosterone enanthate vial label says store at 68-77°F (20-25°C). It does not go in the fridge. If crystals form in the cold, the label says warming and rotating the vial between your palms will redissolve them. The 5 mL multi-dose vial label gives no day count after opening. The Xyosted autoinjector has its own label.
Storage section of the Testosterone Enanthate Injection, USP label (Hikma, 200 mg/mL, 5 mL multiple-dose vial): “Testosterone Enanthate Injection should be stored at 20º to 25ºC (68º to 77ºF).”
Same section: “Warming and rotating the vial between the palms of the hands will redissolve any crystals that may have formed during storage at low temperatures.”
It is in sesame oil with chlorobutanol as a preservative (Description section). Other makers' vials carry their own labels; check yours.
The label gives no in-use period. CDC, citing USP chapter 797, says opened multi-dose vials should be dated and discarded within 28 days unless the manufacturer specifies otherwise. That is a healthcare-setting rule and the nearest primary source; ask your pharmacist what applies to your vial.
Volume and dose math is on the testosterone enanthate page.
Crystals after cold storage are expected, and the label says warming and rotating the vial redissolves them. Before each use, look for particles or discoloration; if the solution does not clear, do not use it and ask a pharmacist.
Vials travel at room temperature. Keep them out of cold car trunks and hot cars.
More in the travel guide.