To answer your questions, based on the patch notes and the official blog:
Will they appear on the wild as shiny pokemons do?
No, only pokemon that are traded can get the lucky trait.
Can all pokemon right now be lucky?
As it looks right now, any pokémon that can be traded can get the lucky trait.
Are there any differences apart from the background effect?
Lucky pokémon require 50 % less stardust to power up than pokémon without the lucky trait.
Will there be a noticeable difference when on the map?
Not applicable since lucky pokémon don't appear in the wild.
Will a lucky pokemon be for everyone or randomly for each user (like shiny)?
Lucky pokémon are a random chance when trading, with a higher chance for pokémon which have been in storage for a while. When a trade triggers the Lucky trait, both pokémon receive the trait.
Source: https://www.polygon.com/pokemon-go/2018/7/23/17602604/pokemon-go-update-lucky-pokemon and the links within.