These are all fascinating Ideas, but I'm not sure they're addressing ...
It's like one third-level spell each round for the entire day. That's not exactly something that can fit in a feat, and it's not well-balanced by gold costs or high requirements.
I would do the following (long to write, but relatively simple to play):
a) Make it a Feat to have an animal companion like this.
b) The Feat wouldn't be tied to a specific animal, but acquiring the animal would need to explained. This might be something as complicated as requiring an Adventure to get a Griffon egg, or as simple as spending gold at The Amazing Emporium during downtime. If the animal gets killed it can be replaced by another one, likewise acquired in whatever is deemed an appropriate manner at the time. The DM (negotiated to an extent with the player) needs to use judgement as to what can be acquired and the methodology appropriate to that creature. Sometimes, the PC might get acquire the animal first and then "train it" --- i.e. get the Feat --- at a later date/level.
c) Out of combat the animal can be instructed to do things. The PC has to make an Animal Handling (Wisdom) roll against a DC set by the DM according to the task complexity. If it is something that the animal would perceive as "risky", then the roll is at Disadvantage. If it is something that the animal would naturally do anyway, then the roll is at Advantage. If the task requires the animal to remember what it is doing over an extended time period, then the animal must make an Intelligence Saving Throw against a DC of 10 to stay on task. DM judgement as to whether this is required and how frequently the saves need to be made.
d) In combat the PC can choose to:
1) Use the animal as a Controlled Mount (following normal criteria and rules).
2) Take an Action to command the animal to attack an enemy. The PC must make an Animal Handling (Wisdom) roll against a DC that is 10 + 5 for every size category that the enemy exceeds the animal. So, if the animal and enemy are both medium the DC is 10, if the animal is medium and enemy is large the DC is 15, etc. If the animal exceeds the size category of the enemy, then the DC is 5. If the CR of the animal is more than double the enemy, then the roll is with Advantage. If the CR of the enemy is more than double that of the animal, then the roll is with Disadvantage. If the roll fails, then the command fails, and the animal may flee (see below). If the roll succeeds, then the animal attacks the enemy (and will take Opportunity Attacks against the enemy) until either the enemy is dead, a new command is issued, or the animal flees (see below).
3) Take a Bonus Action to command the animal to move somewhere within the animal's movement range, with the animal taking a Dash Action if needed. The PC has to make an Animal Handling (Wisdom) roll against DC 10 if the whole movement path for the animal is within line of sight of the PC and the animal. DC 15, if the PC cannot see the whole movement path, or if the animal cannot see the whole movement from the beginning. If the movement will place the animal somewhere it would perceive as "risky", then the roll is at Disadvantage. If the roll fails then the animal may flee.
4) Take a Bonus Action to command the animal to do a Special Move. This requires no roll, and always succeeds. During a Long Rest period (both animal and PC must be taking the Long Rest), the animal can be taught a new Special Move. The animal can remember its Int modifier number of Special Moves (minimum of 1). Teaching a new Special Move might require forgetting an old one. The content of a Special Move is negotiated with the DM, includes things like "play dead", "retrieve an item", "pull a lever".
5) If a command roll is failed, or the animal has less than half hit points and takes any damage, then it makes a Flee roll. This is a Wisdom Save (for the animal) against a DC of 10. If the Save is failed it flees (DM picks where). If fleeing isn't possible for some reason it attacks the nearest enemy.