the 3 legend pokemon's and eggs

As they are legendary they are very hard to catch. Weaken it and give it a status problem and throw a LOT of Ultraballs. To make Pokemon eggs you go to the old couple in Soleceon(I don't remeber how to spell it) and let them raise the same Pokemon, one female and one male or a Pokemon and a Ditto and come back a bit later(you can also check with the poketech app.) and somehow(lol), an egg will appear. Talk to the man outside. Unfortunatly he does not move when an egg is there(like in the Heonn games. As far as I'm aware he doesn't) so the poketech app is very helpful.

Okay, by the three legendary Pokemon, I assume that you are talking about Uxie, Mesprit, and Azelf? Because if you are then I would suggest getting there health down to a low, dark, red colour. Then you need to use a Pokemon that knows a move such as Thunder Wave, Lick, or Hypnosis. If you know one of these moves then you can either Paralyze the Pokemon, or put them to sleep. Then you can try using either Ultra balls to catch them, or use use Dark balls since the Pokemon are in a cave.

In order to make Pokemon eggs you need take two Pokemon of the same species, 1 male and one female, to the Pokemon day care in Solaceon Town. Put them in the day care and give them some time, come back after a while and talk to the old man standing near the Day care's fence, he should say whether the Pokemon can have an egg or not.

Hope this helps.

You can catch the three legnendaries by going into the cave with dusk balls and a Pokemon that knows false swipe and one that can indice a status condition preferably sleep. Then just get them weak and switch to a Pokemon thats as high a level as possible and throw dusk balls.
For mesprit you can use the same tactics except use mean look.

To make Pokemon eggs put two Pokemon of the opposite sex in the daycare in Solaceon town. Then go outside and see what the old man says. If he says "They prefer to play with other Pokemon then they will not make eggs. Anything else and it is just a metter of waiting. Once you get a negg hatch it by keeping it in your party and walking around with it.