Despite the name ground-fault, a GFCI does not actually care about, or use, the ground wire. In fact, GFCI can be added to totally ungrounded circuits in order to provide protection comparable to (in some ways better) that provided by grounding.
With metal conduit and a metal box, you don't need a separate ground wire - the conduit and box are a functional ground.
Since your GFCI/receptacle is installed in a metal box, it should pick up ground via the yoke and a 3-light tester should show a correctly grounded installation.
The previous GFCI may have had too much dirt, paint or (deliberate but stupid) separators between the yoke and the metal box. Or it is possible that, unlike better quality receptacles (which should include all GFCI/receptacles) it simply didn't have the yoke connected to the ground pin.
Your new setup and test results are correct and normal.
As far as:
Always tripping, especially during rain/snow
that is the GFCI doing its job. Rain and snow will cause ground faults on outside receptacles. The solution to that is not to replace the GFCI (though (a) that may have made sense here because of other issues and (b) lots of trips over time are a good reason to replace it to make sure it will work well) but to prevent rain and snow from getting into the circuit:
- Make sure that there are no gaps around the wiring going to the outside receptacles.
- If the outside receptacles are not weather-resistant, replace them with weather-resistant receptacles.
- Add (or replace if cracked/broken) in-use covers to all outside receptacles.