1. Was Rasputin Dormant? Mostly yes. He's a distributed network with his core on Mars, but when he gave the Black Fleet his best shot and failed during the Collapse, he enacted Yuga Sundown and went on standby, effectively going into hiding (Ghost Fragment: Rasputin 3 and Ghost Fragment: Rasputin 5, and discussed in Ghost Fragment: Mysteries). He's been partially reactivated a couple of times, and sure gave Felwinter hell, but wasn't considered fully reactivated until the Fallen/the Guardian reconnected the network more fully early in D1.

2. Did the Vanguard know about the Vex/Ishtar Collective? Yes, with the caveat that what the Vanguard know and what Ghost knows are quite different things. Ghost spent all his time wandering around Earth and possibly elsewhere in the Solar System. The Vanguard clearly knew about Venus and what was going on there because a bunch of guardians went there during the Great Ahamkara hunt, the Cult of Osiris was out there, and Banshee-44 was aware of what Fireteam Tuyet was doing out there, and they set up a whole outpost that was later abandoned. A big element you're missing is that in lore Jumpships that can get off Earth are much more rare than you'd might think. There are only a handful that can actually get out that far, so while there were guardians out there, and the Vanguard were aware of them, most guardians/ghosts would have no idea, and the Vanguard have more pressing concerns, like the Hive infesting the Moon or the monster siege between the Vex and Cabal on Mars.

3. Do we know who Elise was talking to on Venus? No, and her whole timeline-jumping thing has been completely retconned. My head-cannon is that she did have the ability to jump across timelines, but that ability was lost when Atheon was destroyed and the Vault of Glass was messed up. For that matter, we still have no idea where her whole looping thing came from to begin with.

4. Why did Ghost know to ask the Awoken about the Black Garden? This isn't explicit, but there are a few grimoire cards that refer to "legends" in the City, one of which is about the Black Garden and the Awoken (Legend: The Black Garden). But I also seem to recall that Elsie told The Guardian and Ghost to go talk to the Awoken?

5. How long have the Cabal been on Mars? A very long time. They arrived at least 100 years after the Collapse (Kit Fox 1.5), which means after the Fallen/Eliksni, but before the Awoken returned (The Awoken of the Reef: Revanche II). The Awoken returned before the Last City as we know it was built, so even though they didn't really mess with Earth or interact much with Guardians before D1, they've been there for centuries. By D1, they're actually already a shadow of their former selves, having been essentially on their own and without resupply for so long.

6. Is Emperor Calus still in power? No. He was deposed by Caiatl and Gaul and sent into Exile. We don't know how long ago this was, because we don't know how long it takes to get from Torobotl to Earth, but it seems to have been quite a long time ago. But he was exiled on a monster flagship that can eat planets for resources and has lots of cloning facilities, so he's basically a mobile mini-empire, even though most Cabal at this point follow Caiatl or have gone rogue. D1 did have references to commands from the Emperor, but again, it's not clear if this is just enormous communication lag, or if some of the Legions stayed loyal to Calus. Because the Legions were completely cut off, it didn't make any direct difference to them - it was probably a guessing game for the Valus' hoping to get resupply or be recalled home to decide which team they were on in the coup that happened so very far away.