How do I start the quest of saving the Elf Queen?
By progressing through the main story line. It'll be impossible to miss since it's a main quest. I'm going to give you a bit of advice before hand. If you want those dresses, start killing alot of mobs like slimes, since you need a certain item (Blue Tokens iirc) that act out as currency for said outfits. You can only get those tokens by selling monster parts or fishing parts.
Nope, only animation so far is in Chilliste's Card.
How I keep my save file for reintstalling the update or do I just lose my save file?
You need to restart your game. There's been a huge code overhaul in 0.3.1, making 0.3.0 files not compatible.
how do you get mechanical peaces?
In the middle of the night, there'll be a goblin (if you've done the slime slaying sidequest in Hidden Village, which is the starting town) inside the Mountains. He sells those mechanical parts, but he needs you to do a little quest for him first which will later develop into a scene which you MAY consider "Cheating" I guess ? Or NTR, Idk how to differentiate, I don't really care about it either. But yeah, just be warned, Nitt's gonna get it on, and not from the MC.