[RPG] Do you have to take the Attack action before you can make the bonus-action attack from the Crossbow Expert feat

No.

You can't take a bonus action during another action, thus, you cannot take the shield push's push bonus action until after you've used your attack action.

To address the possibility of taking it before you take the attack action, the power seems to specify that it happens because you take the attack action. If somehow, the shield push invalidates your ability to take the attack action (like, perhaps, you push the only available enemy off a cliff with the push), you've invalidated the ability to take the shield push action in the first place.

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.