[RPG] Owl familiar capabilities while executing the Dragon’s Breath cone

Yes, the familiar can use the effect of dragon's breath

The Find Familiar spell description says "A familiar can’t attack, but it can take other actions as normal". The term "Attack" means you're making an attack roll. You don't need to attack in order to use the effect of the Dragon's Breath spell (XGtE, p. 154), because there is no attack roll involved, but a saving throw instead:

You touch one willing creature and imbue it with the power to spew magical energy from its mouth [...] the creature can use an action to exhale energy of the chosen type in a 15-foot cone. Each creature in that area must make a Dexterity saving throw, [...]

Per the rules on making an attack, for something to be considered "an attack" in 5e, it must involve an attack roll or otherwise must be called "an attack" explicitly in the rules (as with grappling and shoving):

If there's ever any question whether something you're doing counts as an attack, the rule is simple: if you're making an attack roll, you're making an attack
(Player's Handbook, page 194)

you can use the Attack action to make a special melee attack, a grapple
(Player's Handbook, page 195)

"An attack involves an attack roll or doing something that the rules call an attack, like grappling or shoving"
(J. Crawford's comment)

See also: What counts as an attack?

Since the familiar isn't making an attack, it can use the effect of the Dragon's Breath spell.

Jeremy Crawford confirms this

As @V2Blast has pointed out, an unofficial tweet from rules designer Jeremy Crawford supports this answer:

Dragon's breath is cast on a willing creature. Your familiar is a creature that can be willing. Find familiar prevents a familiar from attacking, but dragon's breath involves no attack.

No, but it depends on the DM.

The DM can allow this because it is a dramatic move. But one prone to abuse. Because as long as you have superiority dice and acts before the enemy, you could just move an ally.

The RAW order of actions is:

1. In your turn, you ready an action to attack the dragon when it uses breath weapon.

2. In the dragon's turn, it breathes fire. Your friend is toasted.

3. You atack the dragon and the friend (if it is still alive) moves.

This happens in this order because:

When the trigger occurs, you can either take your reaction right after the trigger finishes or ignore the trigger

^{boldface text added by me}

So you cannot interrupt the dragon's breath. You take your readied action after the trigger happens. Your friend will be roasted.