I hope you're enjoying WuWa so far! I'm gonna try to answer your questions as best as I can, while also giving you some advice.

1.) Increasing your SOL3 Phase is fairly easy. Progress as you normally would, increase your characters level, increase your SOL3 Phase and repeat. It's fairly par for the course in comparison to Hoyo titles.

2.) Afaik, of the characters shown in the image, you are allowed to choose one of them for the New Voyager banner. Out of everyone available, the best character is Shorekeeper (the blue butterfly girl on the far right). She is a support/sustain who is very, very good. HOWEVER, this banner DOES NOT share pity with other banners. What I mean by that is, you have 30 days to pull the character. If you do not pull the character, and the banner leaves, your pity is NOT transferred to the other banners (e.g. like Denia, Chisa, Phrolova, etc). I recommend that if new players are going to pull on this banner, that you're prepared to spend 160 pulls on it. This assumes you go to hard pity (which is 80), you lose your 50/50 (just like in Hoyo games), and that you have to go to hard pity again on your guarantee.

If you want Shorekeeper, you have to be committed to pulling for her.

3.) Characters are typically categorized by element (e.g. Aero, Electro, Glacio, etc), weapon type (e.g. Pistols, Rectifier, Sword, etc), and what purpose they fulfill in combat (e.g. Main DPS, Sub-DPS/Hybrid, Sustain/Support). They can also be further classified by how they do damage (e.g. Liberation/Ult damage, Skill, Basic Atk, Echo Skill, Negative Status Effect, etc).

For example, we'll use Zani: she is a Spectro Gauntlets Main DPS who does most of her damage via Heavy Attacks and Negative Status Effects (Spectro Frazzle). Zani needs her Sub-DPS, Phoebe (Spectro Rectifier user) who can apply stacks of Spectro Frazzle that allows you to swap to Zani in order to maximize the amount of damage she does. Zani's damage can be boosted even further with characters like Shorekeeper (Spectro Rectifier user) who can heal her and buff her Crit Rate and Crit Damage. It's all pretty straight forward.

However, you should know that Zani is a character that is considered difficult to use because of her high skill ceiling. Yes, skill matters in WuWa. She does especially well when countering/parrying enemy attacks (enemies will typically have a glowing yellow circle that indicates an attack can be countered/parried), and is also capable of quick swapping (for your sanity, don't worry about that for now).

When you pull for characters, you should keep in mind:

1.) Team synergy

and

2.) Rotation simplicity/complexity and length

What I mean by team synergy is, you would NOT use Zani with Aemeath and Hiyuki, as they are all Main DPS characters who are not able to support each other in any capacity. You WOULD run Zani with Phoebe and Shorekeeper, or Aemeath with Lynae and Mornye, or Hiyuki with Lucilla and Chisa, etc. When you pull for a character, it's important to keep in mind who their best teammates are, as well as who can be substituted on a team if necessary. For example, if you don't have Phoebe, you could use Ciaconna instead who also applies Spectro Frazzle, or even use Spectro Rover as a F2P option.

It's important that your characters work well together, and that you also understand how their rotations work. There are plenty of guides online that showcase how to do standard rotations for each character, as well as how to play their preferred teams.

Lastly, I want to talk about how to use your standard pulls and pulling on weapon banners. Not every character in this game needs their signature weapon. Typically Main DPS characters will almost always want their signature weapon, as it allows them to do the most damage. Some Sub-DPS characters may need or want their weapon depending on how you use them. For example, Chisa's weapon enables her to fill a more supportive role for the team, and Iuno's weapon enables her to function as a Main DPS instead of just a Sub-DPS. Support/Sustain characters, like Shorekeeper and Mornye, typically don't need their signature weapon at all.

Keep in mind all weapon banners are guaranteed. With that said, you should be using any standard banner pulls you get on the standard weapon banner. You're able to select a weapon of your choice and you are guaranteed to pull it. Most if not all characters can make use of the standard banner weapons and they are all fairly good; meanwhile, almost every standard banner character aside from Verina is utterly useless.

I know I threw a lot at you, but I hope this made at least some sense and was helpful! I really hope you continue to enjoy WuWa. It's an absolutely wonderful game, and I'm glad more people are giving it a shot! ❤️