The overarching effect will be that crossbows will become more attractive weapons than they are now, favor combat tactics taking place over intermediate distances, and may edge out some weapons (particularly if wielding two hand crossbows). It's unlikely to seriously unbalance your game.
Crossbows are ranged weapons, meaning that you can deal damage while remaining outside of melee range and so reduce the risk of taking damage yourself for each attack you make. This is similar to any ranged weapon, such as a longbow, with some differences in effective and maximum ranges (30/120 for a hand crossbow, versus 150/600 for a longbow).
However, longbows have the Two-Handed property and so require two hands to use, meaning that most characters will be able to wield only one bow at a time. Hand crossbows do not have that property, and so two could be wielded at once. This excludes light and heavy crossbows, which also have the Two-Handed property (h/t Theik).
This is usually balanced by the Loading property, which limits each crossbow to a single attack per turn. Due to this, even dual-wielding crossbows in the manner suggested in the question would provide an uneven benefit because each attack creates the need to use an action to reload the crossbow for another attack. Even using two crossbows wouldn't overcome the fact that each one can only fire every other turn.
Removing the Loading property undermines that balance at no cost, effectively doubling damage output over the course of combat without any other factors. Now a character wielding a crossbow gets to make a ranged attack every single turn, at minimum, even without Extra Attack, while also keeping a hand free for other purposes (including holding another crossbow, or always having a shield equipped to use). Anything which allows the character to attack using a Reaction or Bonus Action (particularly Crossbow Expert) would mean more attacks. When Extra Attack is available as well damage output increases even more.
Those sorts of options aren't available to any other weapon-- melee requires getting into melee range, which carries additional risk to the character, and "traditional" ranged attackers don't necessarily get to dual wield (longbow), ignore ammunition (javelin), or make multiple attacks per turn (spellcasting). And since hand crossbows deal a base damage ranging from 1d6, you have created a situation in which an attacker engaging over intermediate distances will almost certainly be worse off for using anything but a crossbow. Crossbows go from being one option among several, with advantages and drawbacks, to being an option with no drawbacks while retaining its advantages, even without the Crossbow Expert feat.
Without the Loading property, crossbows become twice as good (if you're only using one) to slightly more than that (if you're using two with Crossbow Expert). It will also make the Crossbow Expert feat slightly less attractive, as it can be used every turn "for free" but would almost certainly have allowed a character to elide the Loading property in the first place.
The specific bullet points in the question:
Holding two crossbows will be possible for any character, not just Rangers. But using them effectively will require the Crossbow Expert feat, as otherwise there is no reason you couldn't use a single crossbow for each available Attack (without Loading, that is). And as written, there is no reason the one-handed weapon attack which triggers Crossbow Expert's crossbow attack via Bonus Action couldn't be the same crossbow (since we're ignoring Loading). Two-Weapon Fighting and Dual Wielder are both only for melee weapons, so they add nothing to this.
No, you don't have to focus on a single target when making additional attacks whether they are on different turns or the same turn through Extra Attack. Crossbow Expert doesn't limit valid targets to those targeted by the first weapon attack. Neither Dual-Wielding nor Two-Weapon Fighting alter this, but those aren't available for ranged weapons and so it's doubly irrelevant.
No, this is definitely not allowed RAW. The rules clearly assign the Loading property to all crossbows, so removing that propery is already not RAW.