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.
Crossbow Expert
- Ignore Loading Quality on Crossbows you are proficient with (This allows for the extra attack).
- No 5ft Disadvantage
- When you use the attack action and attack with a one-handed weapon you can use a bonus action to attack with a loaded hand crossbow you are holding.
So no Heavy Crossbow 3rd attack, nor a hand-crossbow 3rd attack as a heavy crossbow is a 2-handed weapon.
Sharpshooter
- No Disadvantage at long range
- Ranged weapon attacks ignore 1/2 and 3/4 Cover
- Take a -5 penalty to attack roll for +10 damage
This is on par with each of the weapon masteries. You sacrifice accuracy (25% of the D20) for around a 50% increase on damage.
As far as the superiority dice, they're not OP, just spells for that Fighter archetype. They're on par with Paladins' Smite, Rangers' at will Hunter's Mark, and Rogues' Sneak attacks.
According the the DMG level 5-10 is when the group begins to feel its power, so it's natural for them to begin to feel more powerful. Make sure you're using the Encounter calculator on Page 82 of the DMG. This will allow for more control on your part, if you feel the group isn't being challenged enough, step up the difficulty a bit.
Best Answer
Yes: At 5th level, extra attack includes the Longbow
The Fighter, Barbarian, Monk, Ranger and Paladin all get Extra Attacks at fifth level. None of them specify "melee" or "ranged" therefore there is no limitation per the rules as written. All of them read the same way.
Weapons Properties for the Longbow show no restriction
There is no stated restriction on longbow attacks. If the longbow had a special property beyond Heavy, Two-Handed, or Ammunition it would be listed in the Weapons Properties section of the PHB and on the table.
Your DM is not applying the rules as written with that ruling. Getting the arrow out of the quiver is part of the attack, it is not "interaction with an object." The rules are clear on that. (The short bow also has no such restriction).
I recommend that you discuss this with the DM before the next game session - before play begins. Based on the game's mechanics/rules, the longbow does not have the same loading feature as the Hand Crossbow and the Heavy Crossbow. At 5th level, each turn you can attack twice with the Longbow. (My Ranger enjoys doing this now that he's fifth level). If the DM rules otherwise after this discussion, then you'll need to figure out if you can live with that or not.
Excerpted from the Weapons table on page 48 Basic Rules (Same info in PHB):
\begin{array}{r|ccc} \text{} & \text{Martial Ranged Weapons} \\ \hline \text{Crossbow, hand} & \text{1d6 piercing} & \text{light} & \text{__} & \mathbf{\text{loading}} \\ \text{Crossbow, heavy} & \text{1d10 piercing} & \text{heavy} & \text{two-handed} & \mathbf{\text{loading}} \\ \text{Longbow} & \text{1d8 piercing} & \text{heavy} & \text{two-handed} & \text{__} \\ \end{array}
There is no Loading property for Longbow, unlike the two crossbows on that list; thus there is no "once per action" limitation with that weapon.