1). What is the best move set for a Modest Dialga? Timid Palkia? Why? (I need to understand the logic for picking the moves)
The answer to pretty much everything in pokemon is "it depends on your team", but I'll try to go over the basics.
Dialga: Dialga can basically run two types of sets: Defense/Support and Offense/Revenge Kill. A Modest defensive Dialga set and the reasoning behind the moves would look like this:
Dragon Pulse/Draco Meteor - For STAB. Dialga has base 150 SpA, so you should utilize that.
Toxic/Thunder Wave - Toxic hurts pretty much every common Dialga switch-in except Scizor and is great against Stall teams. Thunder Wave is great for aggro teams since they can slow down everything to a pace they can dictate the match with and also hurts common Dialga switch-ins except for Groudon, who is probably the most common. And again, it really depends on what your team needs more.
Stealth Rock - If you don't have another pokemon on your team to set up SR, Dialga does a great job of it. If you have someone else to set up rocks though (Groudon, Deoxys, Mew, Heatran, Jirachi, etc) you can run Roar or a coverage move in this spot.
Flamethrower/Thunder - For coverage. Flamethrower is especially good for taking care of Scizor.
A Modest agressive Dialga set and the reasoning behind the moves would look like this:
Dragon Pulse/Draco Meteor - For STAB.
Flamethrower/Fire Blast - To hit Steels.
Thunder - To hit Kyogre.
Aura Sphere - To hit random stuff like Darkrai and other Dialga. This spot can be replaced by another dragon move though.
Palkia: Unlike Dialga, Palkia can really only run offensive sets and be viable. A typical Palkia set will look like this:
Spacial Rend/Draco Meteor - STAB Dragon attack. Spacial Rend is chosen over Dragon Pulse because of its increased base power and critical hit rate.
Surf - STAB Water attack. Not terribly useful in Ubers with half the pokemon being dragon types, but with Rain support this can still do a lot of damage.
Thunder - To hit Kyogre.
Fire Blast - To hit steel types when the sun is out since then Thunder will miss and Surf will do minor damage. Again, this can be replaced with another Dragon attack.
2). Which items should I be using? Lustrous orb? Adamant Orb?
Dialga: Leftovers for support, though Chesto Berry can work if you want to use it as a lead. Choice sets are obvious for items. If you're playing in a metagame with Arceus, I'd reccommend using Scarf Modest Dialga as a revenge killer against SD Arceus. Dialga's resistance to Extremespeed and Shadow Claw/Force make it an extremely good choice against the SD Arceus sets that do not run Earthquake or Brick Break. Even if he can't hit it for Super Effective damage, he can still hit incredibly hard with Dragon Pulse or Draco Meteor. If you run at least 132 Spe EVs, you'll always outspeed Arceus, even if it has a +Speed nature.
Palkia: It depends on how badly you need Palkia to be choiced. Scarf is Palkia's most common set because it's one of his best, but faking a Scarf using Expert Belt, Lustrous Orb, or Haban Berry can be a good idea too. You could also run Specs I guess, but I can't think of why you'd want to.
3). EV spread? Why? (I need to understand the logic for picking the EV spread)
Dialga: For Scarf Dialga you want 252 SpA/252 Spe to maximize its speed. For other offensive Dialga, you'll probably want something more like 216 HP/252 Spa/40 Spe. This outspeeds most non-RP Groudon, maxes out SpA, and give reasonable bulk. For defensive or support Dialga you want 252 HP and at least 120 SpD for bulk. The rest of the EVs you can play around with - 40 to 60 Spe will help you outrun Groudons and Def EVs will give you more physical bulk. The Pokedex recommends investing in SpA, but I really don't think you need to. Dialga is a powerful enough special attacker as it is and extra SpA EVs seem like overkill for a set that's not offensive. But if you're having trouble 2HKOing Forry in the rain with Flamethrower, you should definitely invest in some SpA (but this shouldn't be a problem).
Palkia: 252 SpA/252 Spe/filler. Palkia is an attacker and you need to maximize his attacking stats.
4). Any other analysis/discussions that may help me.
The Pokedex can explain things better than I can.
For Dialga you need to watch out for Groudon.
For Palkia you need to watch out for Lati@s.
What uber pokemon would go great with a mewtwo in terms of amount of weaknesses covered?
If you're talking about pokemon who resist what Mewtwo is weak to, pretty much any steel type besides Jirachi, Metagross, and Bronzong will do. Dialga, Forry, Heatran, Registeel, Skarm, and Scizor work well.
If you're talking about an offensive pokemon who can kill whatever resists Mewtwo's attacks, then it would probably be easier to just change the set you're using on Mewtwo. Mewtwo has one of the best movepools of any pokemon and he can use just about every powerful special attack there is aside from Draco Meteor.