DUDE THANK YOU SOOO MUCH im doing way better in battles now thank you :)

An easy way to catch a pokemon is to give it a Focus Band and use the move Endeavor to bring its heath to 1 then start throwing your pokeballs.
After use of Endeavor you might want to swap out pokemon if it dosent faint anyway!

(WARNING Focus Band SOMETIMES prevents fainting!!!)

I have found that the best double-battle combo in HGSS would be Mewtwo/Lugia, and preferably at a high level.

Mewtwo:
Psychic
Thunderbolt
Earthquake
Focus Blast

Lugia:
Extrasensory/any powerful psychic move
Dragon Pulse/AncientPower/Hydro Pump
Shadow Ball
AeroBlast

They cover a wide range of eight types and both get at least one STAB. Also, Lugia has awesome defensive stats and Shadow Ball can take care of its nemesis, the Dark/Ghost types. Dragon Pulse gets rid of things like Lance's team and Clair's too, and AeroBlast lashes out at a lot too, especially fighting types. Extrasensory tosses aside lots for the Psychic Pokemon Connection (no pun intended)

Mewtwo is basically an attack/special attack sweeper with great HP. It has the best moves from four types...Thunderbolt for its better accuracy over same-type Thunder attack, and to get rid of all the Water/Ice types out there which are fairly common. Focus Blast lashes out at Dark types and basically kills Normal types like Red's Snorlax as well. Earthquake's great as it's damned powerful and in a combo attack with Lugia, the part-Flying Lugia won't get hit by Earthquake at all, and it looks after steel/water types like the Sinnoh Emperor Pokemon. Psychic's there to get a kick outta Mewtwo's STAB as one of the best psychic moves around. It kills all the poison types and the fighting types as well as others I won't name.

For those legendary-crazed trainers out there, this is the ultimate double-battle combo for you.