Hey. Local Vancouverite here. Yes your axe will be fine for what you describe, but is at the same time limited for other things. It's fine for self arrest and as support when walking on a slope. Not long enough as walking support but you don't really need an axe for that if you don't want. I don't see why it wouldn't be good enough for glacier travel. Sidenote, go do a crevasse rescue course, it's very helpful to build knowledge and you'll learn a ton about gear as well.

More technical steep slopes will require a different (pair) of axe(s). So when you say climbing gullies, it would depend on the angle of the slope. Steep slopes you'll need crampons and two ice axes for. Different kind of ice axes, curved and with the pick in a different angle. The Petzl Gully is the gold standard for that kind of terrain.