Yes, a warlock can keep the hex spell up outside of combat.

In-game, a turn is defined to have a duration of 6 seconds. When not in combat, you are not in initiative order, so turns are not structured rigorously. Nevertheless, your "turn" out of combat occurs whenever you have the opportunity to do something. As long as you have 6 seconds to spare, you can use your bonus action, and therefore you can redirect your hex.

As you noted, this is also implied by the extremely long durations that Hex can have. Nobody will ever be in a combat that lasts 8 hours (that's 4,800 turns!), so the only reason for the duration to be that long is to allow it to be maintained across multiple combats (or applicable social encounters) throughout an extended period. This is also mitigated by the Warlock's spell economy: they have very few slots to spend, so the requirement to spend a slot on Hex mitigates what could be perceived as an overpowering effect.

The damage roll is all the dice together. Add the bonus after.

When you deal damage with a single* distinct attack or other effect, you roll all the indicated damage dice, including any dice added by critical hits or special effects you benefit from like a curse, hex, or sneak attack. That counts as making the damage roll. If you have a feature that lets you add a bonus to the damage roll, that happens after totaling all the dice rolled.

The rules you seek are on page 196 of the Player's Handbook under the Damage Rolls heading. Here is the most relevant passage for this instance:

Each weapon, spell, and harmful monster ability specifies the damage it deals. You roll the damage die or dice, add any modifiers, and apply the damage to your target.

In your example, you will roll two d8's and one d6, add the numbers on those three dice, then add your proficiency bonus (such as +2, for example). The result is the damage you deal.

If you have multiple bonuses due to some combination of effects, they are all processed after the dice are rolled collectively, unless the text for the effect indicates that you add the bonus to each die individually or some other exceptional process.

(*If you're actually making multiple distinct attacks or dealing multiple separate instances of damage such as with a spell with multiple targets damaged separately and not simultaneously, each has its own distinct damage roll. The rules for the attack, spell, or other effect will explain how and when damage is dealt. Basically, whenever the text tells you to deal damage involving dice to a particular target, that's a damage roll for that target. Read the rules under Making an Attack and Casting a Spell in chapters 9 and 10 of the PHB for how to deal with other cases.)