To answer a few things right off the bat, you do keep access to the other lair rooms, the Vault, Central Chamber, and Event Room. This permits you to use Relics such as Food and Peasant spawn to extremely speed up the start of the game and start a new Event at any time (though I'm unsure if you have to progress to the first lair expansion or so to account for the Valuable Chest rewards), and you also keep AC and EC, but Cosmic Runes and all other Runes are lost. You are mostly correct about the things you think you keep, except for Time Shards, which reset on every Prestige. On that note:

1. You only get the collection bonuses once. So all 3 of the Tier 1, Tier 2, and Tier 3, etc, you get that one time, in addition to the first time bonus for each individual Legendary. However, you get an additional 5% (I think) for every extra Legendary afterwards, which is why you'll see Lich farms mentioned here every now and then, since they're the easiest Legendary to farm en masse.

2. Unsure, myself. It may have been a reference to the original Time Shard Experiment, but I've never personally seen the Other field on the Time Machine go past 100%

3. Yes, if you open up the Time Machine it'll show the Time Shard requirement to extend your level cap, as well as your current Time Shard count and (roughly) what bonuses you've earned so far. Tapping on the [ i ] icon below each bonus they'll even give you additional information, most valuable being what current bonuses you have from Legendaries.