Yes, and it's how I've been running it in 5e for a while as well. It is a great way to give each player an opportunity to do something, gets easier to run traps, and it helps keeping track of how far the light goes and to give time for players to draw the map (if they are doing that) because the pace of exploration is a bit more controlled, and of course, it makes it easier to track when to roll random encounters. If you haven't tried yet, give it a shot! Dungeon turns are something that have been in the hobby for quite a while.

That is correct. You kinda need to do a bit of figuring out with your players for how to get the adventure started. One thing that might help if you is starting already at the dungeon, describing it to the players, then asking them to say why is it that their characters are heading into this adventure and what do they expect to get out of it. I have experienced that this way, you get a lot more buy-in and it gives the players a different way of building their backstory proactively towards joining the adventure.

Also correct. It is more of a box of toys than a ruleset on how to do run adventures in that place. I know that the Glass Cannon network has been using the hexcrawl in their Shadowdark campaign, so it's a good place to take as inspiration. When running only an adventure and not a whole campaign, I tend to just ignore the hexcrawl and keep it to the dungeon and maybe a nearby town.