You load as part of any attack with a crossbow. You can draw the ammunition as part of the attack, so loading it is implied to be part of the attack as well. "Loaded" in this case means only that you have ammunition available for the attack. Per the official errata, "loading a one-handed
weapon requires a free hand."
This means that, post-errata, using rules-as-written (RAW), you cannot indefinitely fire two hand crossbows in the same turn. If you have a single attack, you can fire once with your action and once with the second crossbow as your bonus action with Crossbow Expert on the first turn of combat if they start loaded. However, you can't keep dual-wielding because you won't be able to reload with both hands full.
If you stow the second crossbow on the first turn with your free object interaction, you can fire the main crossbow twice on the second turn, since Crossbow Expert does not say the bonus hand crossbow needs to be different than the one-handed weapon that triggers its bonus attack. You may need to load it before the bonus attack using your object interaction, depending on your GM's interpretation of the feat. You can do this on each subsequent turn to fire a single hand crossbow twice per turn (or more than that, if you have Extra Attack).
You didn't request a rules-as-written answer, so I'll note that there is nothing game-breaking about allowing dual-wielding hand crossbows using Crossbow Expert.
Because of the way crossbows work, I think your GM would be perfectly justified in waiving the errata rule in this case but requiring you to get a customized crossbow and ammo belt to allow you to load it one-handed. That would be part of your house rules, though.
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.
Best Answer
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.