I'm trying to optimize the damage done by a ranged Beast Master ranger, and was thinking of choosing the Stirge as a companion.

Blood Drain. Melee Weapon Attack: +5 to hit, 5 (1d4 + 3) piercing damage, and the stirge attaches to the target.

The Stirge will have 12HP (4x my level), 16 AC, +7 attack modifier, and do 7 points of damage per attack (all increasing with level and proficiency). It is not the creature that does most damage per attack, no, but consider the following part of the attack.

While attached, the stirge doesn't attack. Instead, at the start of each of the stirge's turns, the target loses 5 (1d4 + 3) hit points due to blood loss.

Can I command my companion to attack, and on the following turn leave it attached while I attack? This means another 7 auto-hit damage with it, my possible attack, and the stirge doing the Dodge action (per Errata '18 at "Ranger: Ranger's Companion"). The stirge will detach itself after draining 10HP, so I could pull this off every two-turns.

Up to 11th level, when companions can finally take multi-attacks, the Stirge seems a pretty solid companion. After that, probably better companions will exist.

Disgusting, but Legal

Other than the very image of a person with a pet vampire-mosquito-bat-thing, I am aware of nothing in the rules that would prohibit your idea.

This works by RAW

Nothing in RAW will prevent you from taking a Stirge as a companion. You would then be able to use your action to command it to attack exactly as you describe. Once attached, it would not require an attack to deal the damage each round. It would then take the Dodge action each round as per the errata.

I would rule that it can't Dodge

This isn't strictly RAW as the Stirge's stat block doesn't indicate it this way but it makes sense and is how I would run it.

I would consider "attached" to be an equivalent condition to the Grappled condition which states:

A grappled creature's speed becomes 0, and it can't benefit from any bonus to its speed.

Now this does contradict the rules as the stirge states:

The stirge can detach itself by spending 5 feet of its movement.

implying that is has movement to spend. However, I would say that it has 0 speed unless it chooses to detach. If you, and more importantly your DM, agree with this premise then it cannot dodge.

In the text for the Dodge action:

You lose this benefit if you are incapacitated or if your speed drops to 0.

Therefore, the stirge will simply do nothing each turn. Without disadvantage on their attack, it's quite possible your target will simply kill the stirge on their turn.