Yes, you can go in solo and do stuff solo.

The Vengeance campaign is specially designed to have optimized skills in order to reduce calculations on the backend. It's... controversial, but here's a neutral overview:

Each Class's skills have some minor differences to the two morphs on Live; some are more divergent than others (like there are no permanent pets so Sorcerer's pet skills are quite different, but simple skills like the 1H/Shield Bash are basically the same. As such, some builds can be just taken from Live, but some do need more to build around. There are some skill lines like Werewolf of Vampire or Scribed Skills that do not exist. Subclassing is also not allowed in Vengeance.

Armor usually is a non-factor, but there's a test this week where the last bonus of a set (the 'proc' effect from the 5-piece, or the 2-piece of a Monster set, or a Mythic) is active. You need to be wearing that armor to enter the Campaign, and it is also most effective to switch in with the weapons you would like to use already equipped - not all weapons are in your Vengeance inventory.

Food, CP, and Passive Skills do not exist. Instead, there is the Loadout and Perks system that gives boosts to your attributes. As you level up in Veterancy, you also unlock different Perks that function like the Alliance Skill line passives or other similar effects. You also are only allowed to change your setup at the Alliance bases, so you can't swap those in the field.

Builds are something that everyone has their own opinion on. I'm nowhere near a good enough PvPer to give any guidance there; I'm just picking what feels most comfortable for me.

Siege, Potions, and Repair Kits are unlimited, but you can only place one of each siege at a time. You also get unlimited Recall stones but they have a 20 min cooldown.

All quests, Skyshards, and delves are active, but there is a bug with the intro quest if you haven't done it as it requires you to use an item which is unavailable in Vengeance. You'll want to do that in Grey Host or Blackreach, or just skip that quest.

It is possible to go in solo, but Vengeance is a lot more about large-scale combat. Even the NPCs in Vengeance hit harder than they do in Grey Host, so even something like soloing a resource can be a bit of a surprise if you're not expecting it. If you're solo, the best option is to find a zerg and follow them around so you have numbers on your side.

Vengeance is able to support just under 3x the population of Grey Host (a total of 900 players, or 300 per alliance), so it allows more people in. Just having room for that many people though doesn't guarantee that there will be that many people around though.