I collected all 20 eggs, all of them.

But I cant finish the quest at Valenito, he offers 3 quests to bring 5 eggs each, total 15 eggs.

I collected all 20, but there is no star-pattern eggs in my inventory.

Does it mean the eggs are random at pickup and I was unlucky to get enough star-eggs?
What should I do now, abandon the quest entirely?

I've been told that the eggs do not respawn, so there is no way for me to pick up any more star-eggs, means I'm unable to finish the quest?

you go and collect all 20 eggs. Youw ill have now 15 red Eggs (Lost mythical Eggs) and 5 golden Eggs in ur inventory. Make sure you spoke to the BS before talking with Alustin.

Alustin will then have 3 quests. Each wants 5 red eggs and will give u 1 special egg in exchange for it. Open that egg to get rewards. 

What Alustin has to do anything with this event? Quest NPC is Valentino, old shai with blonde beard and a chicken inside it.

It gave me 3 quests, 2 of those I finished, but 3rd is impossible to finish because I already collected and submitted all eggs, and the quest requires me to submit 5 Star Pattern Flondor Goose Eggs, and there seems to be no way to get more eggs because I already collected 20 out of 20 available.

No, we do not.

all 3 quests asks for 5 of the lost mystical eggs and nothing else totalling 15 

do u still have 5 eggs in ur inventory

I have 4 Lost Mystical Egg, which is not required by any quest.

Quest requires 5 Star Pattern Flondor Goose Egg, I have zero of those.

You have misunderstood.  The quest gives a Star Pattern Flondor Goose Egg.  It requires 5x [Event] Lost Mystical Egg, the red ones you've already collected.

You get 1 free [Event] Lost Mystical Egg every day you play at least 60-minutes until April 18.

read the event page, all the quests require 5x lost mystical egg and the rewards are the various eggs that give reward

either you missed one or u turned in one for a cron bundle in which case rip