You can attack three times.
Your first ambiguity isn't really ambiguous at all. The distinction between an Attack action and an attack is pretty clear. Heck, both of the quotes in your question make that distinction.
For reference, the Attack action is defined as:
The most common action to take in combat is the Attack
action, whether you are swinging a sword, firing an
arrow from a bow, or brawling with your fists.
With this action, you make one melee or ranged
attack. See the “Making an Attack” section for the rules
that govern attacks.
Certain features, such as the Extra Attack feature
of the fighter, allow you to make more than one attack
with this action.
Just like the quotes in your question, it refers to the Attack action and an attack as two entirely separate things. So "you can use a bonus action to attack" means that you can use a bonus action to make one attack, not to take the Attack action.
Your second ambiguity is ambiguous; it could mean that each attack with a one-handed weapon allows you to use a bonus action to attack or that the Attack action as a whole does. Fortunately, it doesn't really matter, because:
You can take only one bonus action on your turn, so
you must choose which bonus action to use when you
have more than one available.
So even if every attack you make with a one-handed weapon allows you to make an attack as a bonus action, you can still only take one bonus action and make one attack.
To address your comment, we have the preceding text from the definition of bonus actions:
You can take a bonus
action only when a special ability, spell, or other feature
of the game states that you can do something as a bonus
action. You otherwise don’t have a bonus action to take.
You don't have a bonus action that Crossbow Expert allows you to make attacks with. Crossbow Expert gives you a bonus action consisting of a single attack. Even if it gives you that bonus action twice, (or up to 4 times for high level Fighters), you can still only take it once.
You're missing one thing:
You've used two actions in number 3.
You sheathe your rapier and draw a dagger/knife/dart from some sort of storage. Sheathing the rapier can be done for "free" (first bullet of "Interacting With Objects Around You," PHB p.190). But then drawing the dagger/dart would require the Use an Object action, per "Other Activity on Your Turn," same page. Attacking with the thrown weapon also requires an action, but you're out.
Your proposed scheme had you making six attacks in three rounds. I think you can use the same thinking and still respect the action economy by making one slight alteration. You run into the same draw/sheathe economy problem in a hypothetical fourth round, but the fight's likely over by then. So here's your original scheme, with a small modification:
Load and equip a hand crossbow before combat, along with a rapier in the other hand.
Attack with the rapier, follow up with crossbow as bonus action, without disadvantage in melee range, both due to crossbow expert feat and then move away from the enemy, avoiding opportunity attacks due to the mobile feat. Now sheathe the rapier, for "free".
Draw knife/dart for "free", throw a throwing knife, dart etc., and follow up with crossbow as bonus action, as the crossbow expert feat lets me ignore the loading quality of the crossbow as long as I have a hand free, which I would after throwing the dart/knife.
Fire crossbow as bonus action, still ignoring the loading as I still have a free hand, and then drawing my rapier and attacking, as drawing attacking counts as same action, and then move away, ignoring opportunity attack from mobile feat.
Best Answer
You just need the hand crossbow, and nothing else; the hand crossbow is a one handed weapon, and it enables its own Crossbow Expert bonus action attack. In addition, as Doval points out, PHB errata has removed the word 'loaded' from the Crossbow Expert text quoted in the question.
(Note: I do not take Colossus Slayer into account in the following analysis, as it doesn't particularly affect which weapon setup is the best.)
Turn 1, 2, and 3: Attack action for 2× attack w/ hand crossbow, Crossbow Expert bonus action for 1× attack w/ hand crossbow:
3×(d6+4) = Avg. 22.5 damage/turn, before taking hit chance into account.
This results in a total of 67.5 damage over 3 rounds. Assuming a 60% hit chance, that comes out to a realistic expectation of ~40.5 damage in the first 3 rounds.
If you were willing to expend spell slots, casting Hunter's Mark as your bonus action (instead of attacking) on turn one would bump that up to 88 damage over 3 turns, or 52.8 damage when adjusted for accuracy.
The only way I see to potentially do more damage than that would be something that somewhat violates the spirit of your conditions, if not the word. It also falls behind compared to purely using a hand crossbow as above if the Archery fighting style is chosen rather than Defense.
Carry a supply of rapiers on you in addition to the hand crossbow. For the purpose of this analysis, at least 3.
Assuming you start with the hand crossbow drawn before combat:
Turn 1, 2, and 3: Attack action for 2× attack w/ rapier, Crossbow Expert bonus action for 1× attack with the hand crossbow. Each turn, use your free object interaction to draw a new rapier from one of your excessive number of sheaths, then after finishing the Attack action, drop it for free. You can then reload/fire the hand crossbow as a bonus action since you have a hand free.
This becomes:
(d6+4)+2×(d8+4) = Avg. 24.5 damage/turn.
This results in a total of 73.5 damage over three rounds. Assuming a 60% hit chance, that comes out to a realistic expectation of ~44.1 damage in the first 3 rounds.
If you start with no weapons drawn, the first turn has to be done with hand crossbow only, reducing the damage for that turn back to the same 22.5 as above for a total of 71.5 damage over three rounds (or 42.9 with 60% accuracy).