Your character's Vector rating is the AVERAGE of the Weapon, Support and Prototype that you have equipped. So, add up those 3 Vector numbers and then divide by 3 to get your hero's Vector rating.

In the example below, I equipped items with Vector ratings of 17, 19 and 8. That adds up to 44, which the game divides by 3. The result is about 14.7, but Killsquad doesn't round the averages, so my Vector rating is 14.

No. Vector doesn't work like, say, character levels in ARPGs. Being Vector 15 doesn't make your hero stronger than if he/she is Vector 10, for example.

In fact you may at times have hard decisions on what to equip. For example, you may find your L40 weapon has a couple awesome unlockable passive modifiers, but you may want to equip an L50 weapon with less powerful passive modifiers on it just to get your Vector rating up higher.

It doesn't mean your hero is strong, per se. Gradually increasing your hero's Vector rating via the weapon and equipment used just means you can tackle higher Vector level Contracts comfortably, is how I'd put it.

I think you do gradually find new and cooler weapons and equipment as you move up through the Vector levels.

The Contract Vector level and your character's level affect your damage output and the amount of damage you take.
Generally you'll want to tackle a contract in the vicinity of your character's Vector level. I haven't found the game overly picky on that. Meaning you can take say an L27 Vector character into an L30 Vector Contract and not worry about it.

However, say, an L20 Vector character in an L40 Vector contract would likely not do enough damage to be effective and would take extreme amounts of damage.

On the other hand taking an L40 Vector character into an L10 Contract, for example, you'll do around normal damage and not be an Omnipotent God. :)

For the sake of helping new players, I'd encourage higher Vector players to play some lower Vector levels in the 10s, 20s, 30s etc. New players often struggle to find pick up group partners. You can still have fun playing on much lower Vector level Contracts.

No. Sorry to burst your bubble. :) Rare and better weapons and equipment will have passive modifiers and sometimes active abilities that you can "unlock" at the cost of resources, but these don't raise the item's Vector level.
Just keep in mind you have four heroes to play with. So when you "out Vector" an item's Vector rating, you may find it is still useful to one of your other heroes. :)

A couple places now, and I believe eventually a third place, according to the Dev's Roadmap.

• You may receive a weapon or equipment reward for completing a contract. You'll see tiny icons for these on each Contract "card" on the Contracts page. When you complete the Contract, it'll appear in your inventory. It's not guaranteed that its Vector rating will be higher than what your hero already has equipped.
  
• You may purchase weapons and equipment at the Shop from the main game screen. Both the Contracts and Shops have different tiers. When you get a hero to Vector 35 for example, that will unlock Veteran Mode and "upgrade" the Shop (when that hero is using it) to offer higher Vector items.
  
  
• A late August 2019 update is scheduled to introduce an in-game Vendor from whom you can buy weapons and equipment.

Beyond this, some weapons and items have bonus modifiers that increase the chance of an enemy dropping an item during combat. I'm unclear on if this is actually working in the game currently?

The short answer is, you have to wait for the new Spec. Ops. 3rd difficulty tier/league expected in a late September update and perhaps the new "in-game-session" equipment vendors promised in a late August update.

Just remember that for now Killsquad is in Early Access. Meaning it's not complete, planned features are missing, and some "late game" aspects of the game are incomplete.
For example, right now you can't play contracts higher than Vector 90.

The upcoming 3rd Contracts tier "league," Spec. Ops., is scheduled for release in an upcoming update and expected to cover Contracts in the Vector 120-150 range. That will presumably "unlock" a new Shop tier of some kind.

This does beg the question -- how do players "bridge the gap" between Vector 90 and 120, since currently the game doesn't seem to show a Contracts tier that covers the 90-120 range. Good question, and we don't have an answer yet, as far as I know.