When you make a melee attack with your pact bow, it counts as an Improvised Weapon that you are proficient with, uses Str or Cha (your choice) as the modifier, and all the pact features still apply, but your DM may rule otherwise.
From the Player's Handbook section on Weapons (chapter 5):
If you use a weapon that has the ammunition property to make a melee attack, you treat the weapon as an improvised weapon.
The bow has the Ammunition property and you are making a melee Attack with it.
According to the rule on Improvised Weapons in the same chapter, the bow deals 1d4 damage if you hit:
If a character uses a ranged weapon to make a melee attack, or throws a melee weapon that does not have the thrown property, it also deals 1d4 damage.
You are proficient with your pact weapon when using it this way:
See this passage from the PHB entry on the Warlock (Pact of the Blade):
You are proficient with it while you wield it.
You're wielding your bow even though you're making a melee attack, so you're proficient with it. Note that this specific rule may contradict the general rule that characters are not usually proficient with Improvised Weapons, but may be proficient at the DM's discretion:
Often, an improvised weapon is similar to an actual weapon and can be treated as such. For example, a table leg is akin to a club. At the DM's option, a character proficient with a weapon can use a similar object as if it were that weapon and use his or her proficiency bonus.
In D&D 5e, the specific rule "you are proficient with it", trumps the general rule "at the DM's option". The introduction to the Player's Handbook has a section titled Specific Beats General:
Remember this: If a specific rule contradicts a general rule, the specific rule wins.
You use Str or Cha as the ability modifier:
The ability modifier used for a melee weapon attack is Strength, and the ability modifier used for a ranged weapon attack is Dexterity.
Making an improvised melee attack with the bow counts as a "melee weapon attack". So your attack and damage modifier is Strength. However, since this character is a Hexblade, the Hex Warrior feature applies:
Whenever you finish a long rest, you can touch one weapon that you are proficient with and that lacks the two-handed property. When you attack with that weapon, you can use your Charisma modifier, instead of Strength or Dexterity, for the attack and damage rolls. This benefit lasts until you finish a long rest. If you later gain the Pact of the Blade feature, this benefit extends to every pact weapon you conjure with that feature, no matter the weapon’s type.
So you can use Cha as the modifier, even with the improvised melee attack.
Lifedrinker, Thirsting Blade, and Improved Pact Weapon all apply:
There is no reason that I can see that your eldritch invocations wouldn't apply. E.g. Lifedrinker applies...
When you hit a creature with your pact weapon
That's unambiguous. If your attack hits, you have in fact hit a creature with your pact weapon.
As for Thirsting Blade:
You can attack with your pact weapon twice, instead of once, whenever you take the Attack action on your turn.
That also applies. Ditto for Improved Pact Weapon from XGtE. As the adage goes, "the rules do what they say".
But your DM may rule otherwise:
The DM could rule that if you're using your pact weapon in an improvised way, then it's "no longer your pact weapon" somehow.
See also this tweet from Jeremy Crawford (the lead rules designer for Dungeons & Dragons 5th edition):
If you use a weapon in a way that turns it into an improvised weapon—such as smacking someone with a bow—that weapon has none of its regular properties, unless the DM rules otherwise.
If the DM rules this way, then your pact bow is just an improvised weapon that you're not proficient with, uses Strength as the modifier, and none of the pact features apply.
No.
Or, more specifically, yes, as long as you didn't get those features via multiclassing, which is never.
The multiclassing rules state:
Extra Attack
If you gain the Extra Attack class feature from more than one class, the features don't add together. You can't make more than two attacks with this feature unless it says you do (as the fighter's version of Extra Attack does). Similarly, the warlock's eldritch invocation Thirsting Blade doesn't give you additional attacks if you also have Extra Attack.
Since you only get Extra Attack as a Warlock if you multiclass and you only get Thirsting Blade as a non-Warlock if you multiclass, this rule always applies and you can never use them together.
Furthermore, even if you did find a way to get them both from a single class, you couldn't use them with two-weapon-fighting like that because the bonus action attack from two-weapon fighting isn't an Attack action. You could, however, use them together in that case despite the 'twice' language to get three attacks out of a single attack action without issue: just make your first attack with something other than your pact weapon and then use your pact weapon via Extra Attack and then use it again via Thirsting Blade. Again, though, you can't actually do this because no class has both class features and they don't stack when you multiclass.
Best Answer
With only a pact weapon, you can only make two attacks
Thirsting Blade says:
And Arcane Armament says:
Note that Thirsting Blade does not say that you get to attack "one extra time" with your pact weapon. It says you get to attack twice when you take the Attack action with that weapon. Thus, if you are using only that weapon to make your attacks, you will only be able to attack with it twice per Attack action. Arcane Armament just adds a different way to achieve two attacks - but neither ability will give you a third attack in this situation.
What if you make one attack with a weapon that is not your pact weapon?
Take this example: you made one attack with a non-pact weapon, dropped it then made the second attack of arcane armament with the pact weapon.1
In this case, it actually creates a slightly ambiguous situation if you read it strictly by RAW.
By RAW, Thirsting Blade allows you to attack twice with your pact weapon specifically, and Arcane Armament allows you to attack twice in general; both effects trigger off of the above situation. If an ability is triggered, you should be able to benefit from its effects unless they are prevented by something else. The only way for both of those abilities' effect descriptions to be accurate would technically be to allow another attack with the pact weapon. And there is seemingly no rule preventing this from happening.
However, this is almost certainly not the intent at all. The fact that Thirsting Blade explicitly was added as an exception to Extra Attack (which is almost identical to Arcane Armament) tells us that the kind of interaction is not intended. Note that the artificer is in UA, and not all UA is balanced or tweaked for multiclassing. This is likely just a case of a loophole they haven't gotten around to addressing or noticing yet, but I'd expect it to be fixed upon official release.
1 - Two-weapon fighting would not work here, however, since the bonus-action attack is taken after the Attack action and not as part of it.