I want to challenge your approach to this. As you note, the Light property is defined like this:
Light. A light weapon is small and easy to handle, making it ideal for use when fighting with two weapons.
This property isn't just there as a computer function — it's also part of the basic description. A hand crossbow is small and easy to handle. And it is, as far as crossbows go, ideal for fighting with two weapons. You may need special practice and training (as represented by the feat) to take advantage of it, but it's undeniably better than any other crossbow for that case.
Is it true that mechanically there are no current rules which make particular use of this property? Meh. Probably so. But it does not mean the property shouldn't be there, because that kind of mechanical interaction isn't the only reason weapons have descriptions.
It might have been more elegant from a "the puzzle pieces must fit together!" perspective if Crossbow Expert said "a crossbow with the light property" instead of "hand crossbow", but I don't think that really would add much of actual value.
Of course, there's always the possibility of future features which apply to "light ranged weapons" or the like.
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
Yes, Crossbow Expert allows you to pistol whip with a hand crossbow and then do the same thing as a bonus action
The first part of this interaction hinges on whether or not an improvised weapon is considered to be a weapon for the purposes of rules and features that depend on you doing something with a weapon. We already have a great question and answer about that here: Are Improvised Weapons used in melee actually melee weapons?
As NautArch's answer clarifies, improvised weapons don't strictly get to call themselves melee weapons or ranged weapons, since they exist outside that classification, and therefore aren't compatible with rules that refer specifically to melee weapons or ranged weapons - but they are still weapons at the moment you use them to attack, and luckily Crossbow Expert only specifies an attack with a one-handed weapon, not a melee weapon or a ranged weapon.
It seems obvious therefore that an improvised weapon wielded in one hand - lacking as it does the two-handed property - is a one-handed weapon, and so therefore taking the Attack action and making an attack with it satisfies the criteria which allows you to make a bonus action attack with a hand crossbow. That the hand crossbow is itself the improvised weapon you used in the first attack doesn't matter, since it's well established that shooting a hand crossbow also qualifies to let you make a bonus attack with the same hand crossbow.
Having established we get to make the bonus attack, the second question is whether or not the bonus attack could be used to make another melee attack with the hand crossbow as improvised weapon rather than having to shoot the crossbow normally.
Fortunately the language used in the feat doesn't seem to rule this out at all; it doesn't specify that you "shoot" the hand crossbow, or specify that you must use the crossbow to make a ranged attack - it simply allows that you can make an attack with the hand crossbow. The natural understanding of using the crossbow as an improvised club is that you are making an attack with the crossbow, even if it's not the one the crossbow was designed to make, so there is no conflict between the text of the rules and the action you want to undertake here.
TL;DR yes - attacking with a one-handed improvised weapon satisfies the criteria for Crossbow Expert's bonus attack, and the rules text for the bonus attack doesn't exclude using a hand crossbow as an improvised weapon in melee instead of shooting it normally.