Eldritch Knight is sort of a weird bird. It's a prestige class for making a wizard/sorcerer more tanky in melee. The problem is, you don't want to (ever) cast spells in melee. Doing so provokes attacks of opportunity. Further, without specific feats (Arcane Armor Training, I think its called), you can't wear armor without having a percentage chance (that goes up with the weight of the armor) that your spell fizzles.
What's your goal for the character? To dive into melee and hit people with swords while occasionally blasting them with magic? You might want to look at the Magus then. Its built a bit more around doing so.
To answer your question - the ability to cast 3rd level spells is based on the class's casting progression. The
Sorcerer is a good one to look at since your Charisma is better then your int. In either case (Wizard or Sorcerer), you need to be level 5
in that class to meet the requirement. This means that at level 7, you can enter the Eldritch Knight prestige class. This is consistent for when you can enter most prestige classes.
I've finished through Act I, and I'm taking a break while they come out bug fixes before going further. My experience is that some of the custom companions are workable, others I'm not too fond of.
Valerie - Because the AI actually does lock onto someone and sticks, sending her in first is decent as a 'tank', something that doesn't happen on tabletop. That said, Pathfinder is still a game of rocket tag, and you're far better off killing things fast rather then tanking.
Amari - give her a Greatsword instead of that Large Bastard Sword and let her rage and go to town. I'm sort of wishing I gamed it a bit and took a level of Vivisectionist Alchemist so she could self buff, but pure Barbarian raging works well.
Linzi - I just go straight bard with her. She's a support character, but she's a decent backup healer and her singing makes everyone else do better. With a party of six, you have room for that.
Octavia - Rogue/Wizard, wants to go Arcane Trickster prestige class. I screwed up her build by going Rogue 3/Wizard 2, when I could have gone Rogue 1/Wizard 4 and used a feat to get her sneak attack prerequisite. I'm likely to reroll because I'm not enjoying my Paladin fully, and I'll be building her differently in the future. That said, she's a good skill monkey to have around.
Tristian - He's a cleric 'wizard' in that he stands in the back in his robes, prays nothing tries to hit him, and plinks away with his bow for piddling damage. That said, his spells can really turn around a battle if he lands a Hold Person at a strategic time, and after combat, his Channel Positive Energy to heal up the party is bonkers. Tristian is a better in combat healer because he has the charisma to take a feat 'Selective Channeling' that prevents Channel Postive Energy from also healing the enemy.
Harrim - Another out of combat healer. Heavily armored and can help hold the line. Take Harrim or Tristian, depending on how many bodies you need in melee.
Jaethal - Haven't used her. Not fond of how many fiddly bits Inquisitors get, it makes me feel like you have to focus on micromanaging her. She's also undead, which is a huge problem for healing. She's also Evil, and more towards the Stupid Evil side from what dialogue I've had with her.
Regongar - Chaotic Evil. Half-orc Magus, so he's trying to steal your role. I've read that he's also a micromanagement case, but can be really effective. Doesn't come across as Evil when I've talked to him so far, besides wanting to take bloody vengeance on the people who enslaved him.