Cathamber / Arc Zeekhad

sure. I will get you the specifics on animal companions

The idea is to use what I stayed at guidelines and just pick any 3 items that you think are reasonable. Creation your own story and content.
The rolls are just used to indicate above our below you characters baseline. You can even narrate you own roll results.

The idea is the tower evaluation from the GM/player perspective is to express your characters abilities and methods to me and the other players. For the story it is just an hour of testing, done, move on

With that background he works be considered well learned with math and records so knows how to read but had had limited access to books. Equivalent to at a primary school graduate. History would be fine but book on theory of magic and life would be hard and or slow to comprehend

Technically you do not need to by the very simple system mechanics. However it does fit a story nicely. You could get another trait pertaining to this. Something like one of the basics (below) or a hybrid of them

Dungeoneer: You gain Advantage when attempting to find your way through a dungeon or cave system and when attempting to identify creatures native to dungeons or caves.

Educated: You gain Advantage when checking to see if you know specific information.

Survivalist: These berries are safe to eat… I think. You gain Advantage when Testing to forage for food, find water, seek shelter, or create shelter in the wild.

As for companions, I think of them as typically summoned and not found. I will get you the templates but do not see a problem with the basic ideas you proposed

Animal Companions
Animal Companions are the providence of Archdruids, powerful creatures that follow the call of nature, and serve alongside as friends and allies.
All animals have the same Animal Traits, having a basic melee attack, and a template from the list below. They also get two Traits from the Player Trait List that can make some level of sense when applied to animals. Animal Companions are exceptionally unique and should be different than a standard of their kind, but they should not have limited access to magic, or other Adventurer-specific abilities. Your Game Master has the final say on whether a Trait is appropriate or not. I allow some access to magic but will have to approve it.

Small Creature Template
Little, fast, easier to maintain, this type represents small cats, dogs, or birds.
4 Hit Points
Small Creatures get the Evasive and Sneaky or Fly and Evasive. Add 2 other traits with GM approval.
• Evasive: Whenever you are attacked, you may roll a d6. On a success, the attack misses. This does not stack with Evade.
• Fly: This creature can fly up to 60 feet an action.
• Sneaky: As per the Player Trait (see Page 36).

Medium Creature Template
A medium creature represents an animal such as a medium hunting cat (like a leopard or bobcat), a wolf, a deer, or some other form of animal a little smaller, to a little bigger than human.
6 Hit Points
• Maul: As one action, this creature can make three attacks with Disadvantage.
Add 2 other traits with GM approval.

Large Creature Template
A large creature represents horses, stags, large hunting cats (like tigers or lions), or bears.
8 Hit Points
• Crushing Blow: All attacks this creature makes deal 2 damage.
Add 2 other traits with GM approval.

Arc currently is currently a Novice so with your Quick Shot trait you are good to go. Once you are a veteran you will gain weapon Mastery, which happens quickly but still requires some rest, IE not new skills or trait in the middle of a fight. Both those traits are fine for a companion. FYI : I only allow companions to have 1 mystic (magic/special) trait.
Also If a companion dies you can not simply re-summon it but don't need a full trait to restore it. Either an expensive scroll/magic or a short quest to find/restore the companions physical vesel.

sounds good

Let just say you spend some time training with the guards and other adventurers to improve your weapon technique. That combines with your experience with the tower is enough to raise you to veteran so go ahead and add weapon mastery.

I use notes and private for information to players that others may not be aware of. Players use it the talk with me or other DM without revealing plans and thoughts. Sometimes these are made public later when relevant