I went through the cult cave some time ago, but there is no Oblivion gate near Fort Sutch. I've checked several times so far. Is this a glitch?

Have you started the main quest?

Some gates spawn at random. Maybe it just hasn′t spawned there yet

Sutch gate is supposed to spawn at the same time as the city gates for the Allies quest, so it's definitely an anomaly. Are any of the city gates missing?

Well, maybe he hasn′t arrived at that quest yet. Or is it an early quest maybe?

It should appear after you've finished the Dagon Shrine quest.

Is there any way you can "do the cave" without completing the quest? E.g. if you go to Lake Arrius Cavern before you've done the quest that tells you where to look? I know you can back there later and the generic cultists will have respawned, but I don't recall what the situation is if you go there too soon.

I havent tried, but I imagine that Lake Arrius Caverns would be inaccessible until you start the quest.