I think there are several ways of handling any of these bits; as an old-school style adventure, the goal is to give the GM a lot of agency in how the adventure is run. In any case, this is how I would personally run it!
1. I would have the existence of the dragon-slaying artifacts be made explicitly clear to the PCs in-universe. Maybe when they receive the dragon-slaying sword. Alternatively, a proud Cirothe could send kobolds to egg the PCs on ("I'll bet you can't get the awesome magic tools, heroes. Neener neener!").
2. That's definitely the largest weakness of the hexcrawl format. If this were a plot-heavier adventure, there would probably be a reason made explicit in text. But in the end, it's the question of why Bowser just keeps Princess Peach in the castle. As much as I personally dislike the trope, the kids and the damsel are in distress solely to be motivation to the players. If you want it to make sense, though, I would make the captives vital sacrifices to a ritual that only happens when the planets align (etc.), and that date is quite a ways away. This has the added benefit of putting the PCs on a clock. It's a long clock, but a clock nonetheless.
3. My two ways of dealing with this are:
a. The Schrodinger's Unicorn approach. Call for a Perception check in every hex when they start searching for the unicorn. If they succeed, it's in that hex. If it's not, search another one. Pretty gamist and non-interactive, but it's sufficient.
b. The Just Tell 'Em approach. Dispose with the Perception checks entirely and just tell them it's in that hex (or within one of four adjacent hexes?) and have them check it out. When they make it to the right hex, run the appropriate encounter.