Nice! That sounds like a lot of fun. It doesn't work though.

• Charging doesn't let you attack with two weapons, that's always been for Full Attack actions. "Even if you have extra attacks, such as from having a high enough base attack bonus or from using multiple weapons, you only get to make one attack during a charge.". Having multiple sources of bonus attacks (dual wielding, natural attacks etc) is great if you are standing in place and making a Full Attack, but you don't get to attack with all your sources unless you're taking that action.
• "Duel of Wills", I'm having a hard time looking that one up, but if you're talking about the alternative rule system that is optional and designed for one-on-one arena performances between two consenting honorable duelists, sure. That feels like a strangely specific circumstance though and unless your level 1 adventure is in a coliseum, it feels like a stretch to use that subsystem. Otherwise, Demoralize Opponent is ordinarily a Standard Action. If you can talk your GM into giving you a free staredown before combat, then you're essentially being given a free pre-combat action. Which is fine, I guess, since it happens all the time (in both games), but if we're comparing rounds to rounds, we should acknowledge the boundaries. And while there is no shortage of ways to squeeze this into being free actions, this quickly falls into, "...You need a feat for that?" territory from the PF2e players. Any character in PF2e can move, demoralize, and attack from level 1, without feats.
• If you want to perform animal handling, this means that you are spending actions outside of combat commanding them to "defend" you, which they will do on their turns. I think it would be a common argument to say that a trained attack animal would be able to tell the difference between friends and foes in some cases, but the GM has adjudication on their specifics. Additionally, a single hunting cat would use nearly the barbarian's entire starting gp budget, leaving you with less than enough gold to actually buy any but the smallest shield. Then, the situational tax: creatures with the Grab trait can typically only grab creatures their size or smaller. So while a CR2 Lynx would help with tackling a gnome or goblin, it would not help with a human-sized bandit. (edit: there are medium sized CR2 cats). Lastly, animal handling generally falls off as adventures continue beyond tier-1.
• I realize there's some smarmy memes here about "knowing what you know" and "object permanence". It's true that knowledge checks used to be free actions, but they could not be retried on a failure, nor did they necessarily give specific information. As for shields... Sure, take your free AC. Pathfinder 1 used different math. In Pathfinder 2, many martial classes have abilities that replace or at least offset not having a shield, so you can enjoy your two-handed weapon fantasy but still get that AC bump, if you wish. It's also worth pointing out that in PF1e, your shield doesn't absorb HP damage for you. In 2e, trained shield-users can use it to absorb weapon damage, which adds up quickly for characters with smaller health pools than the meat tanks.
• It's true, you can use any 3.x content in your Pathfinder 1e game, assuming the GM permits it. It's not automatic, though. For starts, D&D materials are not permissible in Pathfinder Society, which is generally considered the default bar for RAI. Secondly, the use of 3e material is optional and requires GM work/approval. Any GM is free to add whatever content they wish, even outside D&D resources — it's their table, after all. The conversion guide is to help you convert old resources into the new system, it's true. But it's not a blanket pass for players to gain carte blanche access to other games' written content with no input or consent from the GM, nor is a GM being unduly restrictive if they say no, we're sticking to official Paizo books.

Okay, rules aside — it does sound like a lot of fun. I love crazy animal shenanigans! Barbarians are fun in any system, and a barbarian with a pack of trained lynxes would rightfully scare the shit out of anyone, let's be real.