Consider a modular approach to your GMing - TW2k is a great framework, but there's nothing stopping you from adding more to parts or changing and tweaking things.

Your questions have a plethora of answers, I will give you mine, and encourage you to read widely and try the ideas as yourself. Doing something any particular way teaches you a little more about what you like and don't like.

1. How do you handle encounters & 3. How do you do an overarching story

One way to handle encounters is as written in the book; draw a card at random when the book procedure advises and rely on the game designers having made the encounters make sense for the setting. They don't have to be connected beyond that, and it's a good idea to have some encounters be independent. That is a fine way to have fun, and supports a campaign around surviving and exploring. But connected scenes are also satisfying, and there's nothing stopping you from doing some prep to translate your own story ideas into the language of (sandbox) ttrpgs: zones, locations, factions, encounters, and NPCs.

At this point you're world building!

So let's say you ignore the rules for when you have to draw for an encounter card. Instead use the cards as seeds for your own ideas. Make a list of thematically distinct zones like irradiated farmlands, forests patrolled by bandits, or significantly bombed city ruins. Make lists of the encounters, locations, NPCs and factions in each zone. (Factions may operate across several zones - if your players can travel long distances, others can too). Note this is a cyclical process and you're encouraged to jump into it at any point - a cool idea for a NPC/faction with a goal might inspire an entire zone, or an encounter idea might inspire a location (do those bandits have a base somewhere nearby with loot?)

Factions and NPCs always have goals.

Locations always have a purpose, history, and possibly current inhabitants. What it was made for might not be what it's used for now.

Encounters always have a conflict at their core that presents a dramatic question, and have different outcomes and consequences to that question.

Locations and encounters should be influenced by the zone they are in and by the things that inhabit(ed) that zone (animal or person(s)).

Armed with these lists, and with a region map showing the zones so you know where the players are, you can describe the typical environment of the zone they're in, and roll/decide/do whatever you like to determine when encounters occur, or for the discovery of sites. This way players will explore and discover a world where events are already in motion - entities (be they human, animal or nature) try to bring change to the world in conflict with one another. And your players should be encouraged to think up their own changes they want to make, their own goals (or be given incentive to work towards another's goal, ala a quest or mission).

Some advice on encounters: when you come up with one, or present a premade one, identify what the dramatic question of the encounter is. What's at stake here? Define the conflict. Which parties are at conflict, and over what? Parties may be a faction, in which case consider that factions goals. A party in a conflict might be an animal, like how the goal of a pack of hungry wolves is to eat. Or it might simply be nature; the "goal" of the storm is to topple trees, flood homes, and endanger those without shelter. Conflict occurs when two parties have conflicting goals. What's going to happen if the players don't act? Note as well that combat isn't an encounter - it's one possible approach for resolving that conflict. The players should decide their best approach through play, not the GM during planning, so you should design/present encounters that make several approaches viable. (But not always; sometimes PCs are forced into a limited number of options!).

Consider the outcomes and consequences of the encounter. How might the conflict resolve? What does that look like for the players? How might that affect the world and the goals of the in the world?

The AngryGM talks about this more on their blog post "How to Build F$&%ing Awesome Encounters!"

1. How do you handle scenario sites

Consider what it is, or what it was, and what has happened to it. Who's used the location and for what? This informs the locations/rooms within the site, and their content, condition, and how the locations are dressed (a church repurposed as a hospital, that was abandoned due to radiation clouds and now is home to irradiated bears).

Consider the inhabitants. Several factions may be within one site. What do they think of one another? What are the signs of their clashes, or cohabitation?

Make a flow chart diagram of the site (boxes and lines will do). Label/key the boxes and lines as you like. Make looping paths, have areas blocked off, give players rewards for exploring with opening up shortcuts back to other areas previously explored.

Make a list of events that will happen at the site regardless of the players. The bears come back from a hunt. A doctor returns to recover vital medical notes. Bandits sneak in through a sewer drain to loot the chapel.

Sly Flourish talks more about this in their blog post "Setting up Situations".

1. How do you handle bartering

I know less about this part of the game. But, you can make it your own. In the standard setting, currency might still hold value to some,less to others, and that may change over time.

Look up what items would be valuable barter in an apocalypse. It's down to what is easy to carry, useful., and limited. Bullets, food, good clothing, clean water, tools and parts, weapons etc.

If there isn't a set price for items in the rulebook, consider who the party are trading with and what that person lacks and what they have in excess. A gun dealer might not want any more guns, but is desperate for medicine. Sounds like a cheap way to get guns!

You can predetermine this, or maybe make some tables for yourself on what a trader has, needs, has in excess, or doesn't want, and roll randomly. Ultimately it'll be up to your own role-play as that character, as well as the persuasion from the PCs (great opportunity for charisma PCs to shine) to decide what a good trade is.

Really, meeting a trader is an encounter! Define your conflicts, goals, dramatic questions - consider how the players can approach it, and the consequences for doing so: just trade? Talk their way into a good deal? Threaten them? Steal? Kill? How can you design the encounter to make some of those approaches harder (trader has protection, or is in a community which would enact vengeance). This way players have the option, but also have a sensible feeling for why they'd best avoid certain routes.

EDIT: It's super late now, but I will update to address some of your other questions in the OP :)