All of those things have just about simple answers. Ash in Lavitar's dream, was the work of Unown, who hit Misty on the head. They wanted to know the reason why Lavitar always cringed in fear, and Unown witness it so it showed it to them by letting the trio go into Lavitar's dream.
Mewtwo said back in Mewtwo Returns that he wasn't going to erase their memories like he did before. Meaning, Ash, Brock and Misty all remember him still.
In Mastermind of Mirage Pokemon, If I remember correctly Mew showed itself to Ash and Misty first, but I think it was Max who spotted it before Misty's battle. Then everyone else later. They even mention at the end of it, that the Mew had a true heart like a living breathing Pokemon. So that one could possibly have just the same thing as only coming to those with a pure heart.
Like ShinjisLover said, all his friends would sacrifice their lives for their pokemon just like in Lucario and the Mystery of Mew. Max wouldn't let anyone hurt Jirachi after it was proven that Butler was the bad guy nor would May with Manaphy. Misty almost sacrifice her life for her Gyrados. (Can't think of nothing for Brock and Dawn at the moment....)
The father thing is still hard to go after though. In the original it was stated that it was his grandfather, but English dub changed it to his father.
The Psychic connection...umm hello Jirachi chose who as his friend? Not Ash, but Max. Not so chosen one after all is he? What about that Dexoys? Max too...now hold on is Max an Acreus too? (jk thats off topic)
And the Thunder-Armor? Dawn came up with her own move, Ice Jet, but of course Ash was the one to finally make it work with Buizel after he obtained it.
Sometimes half the time in some episodes, they don't even remember what happened in a movie. But of course the movies are non-canon. And why wouldn't Ash see so many Legendary Pokemon? That would make the show boring if we just saw so many Legendary Pokemon popping up, but never get to see any appear. That'd be lame.
Ash didn't remember Mewtwo at all at the beginning of the second movie. He thought he saw something at the end of the first movie which was Mew in the clouds. Of course Misty and Brock said they didn't see anything, which made Ash start to believe the same too. BUT they did all see Mewtwo escaping from Giovanni back near the end of Season 1. And no Legendary pokemon stay together, they all like to be solitary. Most of them that come together end up causing destruction in their path. Only the mother Lugia and her baby Lugia Silver has proven this different, but I'm sure there's no way their together now. They've probably went seperate ways.
I think this is all I can say at the moment, but if I think of more I shall return!:rambo: