Welcome!
1). Pretty much, yeah. There are a couple of god abilities that can swap out gods, but I think it's just lesser gods IIRC. And that ability doesn't come up much, because it's specific to one god.
2). There are 4 greater and lesser gods for each of the factions: so 16 gods in all, between the two factions. Two of them don't show up until you win your first game and unlock trade. But all the rest are available at any time, and it's just luck of the draw which ones show up when.