yes just go 2 the beach & lapras will be there..veronica
yes, in miracle sea or surrounded sea.
after you beat brine cave within the story line lapras appears, I'm pretty sure you can get a lapras (you can't get that specifec one) in miracle sea (or surrounded sea, can't remember)
You can't get Primal Dialga, but you can get normal Dialga after you beat the cave (Lapras will always be at the beach and you can ride him to go to Hidden Land any time)
I think Lapras will come back to the beach area and you can talk to her to go, but I haven't played the game in a wile...
You have to complete the game fully, including the post-game plot and unlocking the Marine Resort dungeon. Spacial Rift will also reopen, it will not mention this, but it will now be on your location list.
He's in the same place you left him, at the top of Temporal Tower. Go to the beach to speak to Lapras to take you back to the Hidden Land. You have to navigate through all the same levels as you did during the first time you came, but Dialga's recruit rate is 100% at least.
No, you can be one in explorers of the time, though.
Talk to Lapras at the beach, and you can go back to the Hidden Land. Dialga is still at the top of Temporal Tower, though you need to go through all the intervening locations again. However once you defeat him that second time, he'll join automatically.
Defeat him at spacial rift then go back there and beat him a second time he will join your team probably.
If you want to get the level 20 lapras in the basement union cave, it can only be found on fridays.
Arceus is not in Explorers of Time or Explorers of Darkness. There is an Arceus statue in Explorers of Sky, but there is no real Arceus that will join your team.
In Pokemon Mystery Dungeon Explorers of Time it is impossible to get Mewtwo. You are only able to get it in Explorers of Darkness.