I haven't played Explorers of Time, but in Sky you find Lugia in the Surrounded Sea. But first, you need this:
1. Get the Secret Slab/ Mystery Part
2. If you have a fire-type hero, get it to about level 75 and make sure it knows a normal-type move
3. I suggest, instead of doing step number two, make the team leader either a Dark- or Electric- type. Lugia is Flying-Psychic and around level 48.
4. Switch your team members' moves off so you can fight Lugia on your own
5. Bring either the Golden Mask or Amber Tear(promotes recruitment)
If I remember correctly, Lugia is on B10F. If you fail to recruit it, keep trying. It should have a high recruitment rate; I got it the first try.
Yes
he is a fake
they ask you if they can be on the tam
Nope
you recruit him by going to electric plateau
If you're talking about Pokemon Mystery Dungeon Explorers of Time or Pokemon Mystery Dungeon Explorers of Darkness, you can recruit as many Pokemon as you want. If you are talking about Pokemon Mysatery Dungeon Blue Rescue Team or Pokemon Mystery Dungeon Red Rescue Team, you can only recruit a certain number of Pokemon per friend area.
No, there isn't.
mission recruit or egg
in the second cave
no but you will find the statue of arceus
dungeon leader can be any Pokemon you recruit. you will be able to change leading Pokemon after you graduate .
concealed ruins with secret slab or mystery part