You need a free hand to load a hand crossbow, but you can attack with any one-handed weapon to trigger
You are correct in that you need a free hand to reload your hand crossbow, which makes this part of crossbow master the easiest if you use a hand crossbow and nothing else, because triggering it with a hand crossbow means you can reload and fire again with your bonus action. But you are also perfectly capable of starting combat with a loaded hand crossbow, attacking with a sword and then using your bonus action to attack with the hand crossbow.
You'll then require a free hand to continue this strategy in the next round, so unless you have a magical self-loading hand crossbow, you'll indeed struggle to use a one-handed melee weapon and a hand crossbow for the entire duration of combat, unless you do some weird weapon dance or show up to the fight with a whole belt of preloaded hand crossbows.
This is most likely the one that is the safest assuming you can't have a dozen preloaded hand crossbows on your person:
- Start with loaded crossbow. If your DM rules that this is not a thing, skip to round 2.
- Round 1: Attack with sword, bonus action hand crossbow.
- Round 2: Attack with sword, free object interaction to sheath sword, bonus action hand crossbow now that you have a free hand.
- Round 3: Free object interaction to draw sword, attack with it, drop it (free action), bonus action hand crossbow now that you have a free hand.
- Round 4 onwards: Repeat round 3, but instead of using free object interaction to draw it, you'll use it to pick it up.
1) Ammunition property, load as part of an attack, would mean you are
attacking twice with one hand crossbow, not duel wielding hand
Not necessarily, you could have two loaded hand crossbows before the fight starts, and then attack with one, and use your bonus action with the other. However, you would be unable to reload them until you have a free hand. However, mechanically speaking, you are better of using one hand crossbow anyway, because it means you require fewer magical weapons. Ask your DM if he's okay with re-fluffing a single hand crossbow into a "brace of hand crossbows".
2) Timing: Do you have to attack with the "one-handed weapon" before
the bonus action crossbow attack since it is the triggering condition?
Yes, you get a bonus action for attacking with a one-handed weapon, so you can't use the bonus action until you've used the main attack that triggers it.
3) One-handed weapon would not include unarmed strikes.
Correct, because that is not a one-handed weapon.
I've houseruled in the past that as part of being a crossbow expert, you can ignore the free hand requirement for reloading the hand crossbow as long as you're wielding a light weapon in your main hand. This worked perfectly fine for our swashbuckler who wanted to look cool. The bonus attack with a hand crossbow is hardly the strongest thing in existence, so penalizing somebody with having to do a weird weapon dance and potentially losing their melee weapon seems kind of unnecessary.