Trade from Fire Red or Leaf Green. In those versions you need to pick bulbasaur as your starter and after you beat the elite four he will wander around in kanto (not on 1/2/3/4/5/6/7 islands) and you can catch him, every time you change route so will he so its up to luck but if you find him he will flee on his first turn so you should probably have a master ball, as for Suicune and Raikou you pick Charmander for Suicune and Squirtle for Raikou (its basically whatever your starter is weak to)
Hope this helps
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No you cannot catch Entei in Pokemon Emerald.
No, you can catch Entei without catching Moltres.
How do you catch latios in Pokemon sapphire
There are many Pokemon you catch that are similar in sapphire and emerald but there are many Pokemon emerald can only get and not sapphire.
Entei, Suicune, and Raikou are not available in Pokemon Emerald, Ruby or Sapphire. They can be brought into the game by trading.