That depends on what you have done before. Nothing special happens after you catch rayquaza. If you haven't done it yet, you can catch Latios/Latias.
No, there is no such thing as a 'Cheese Orb'. However, there are three orbs: the Red Orb (for catching Groudon in Soul Silver), the Blue Orb (for catching Kyogre in Heart Gold), and the Green Orb (for catching Rayquaza in both games).
You do not need to show Rayquaza to anyone in HeartGold. In order to get Rayquaza, you need to show Kyogre and Groudon (from HeartGold and SoulSilver) to Professor Oak. In turn, he will give you the Green Orb. With the Green Orb in hand, you can find Rayquaza in the Embedded Tower.
you continue catching the legendaries if you haven't already got them.
a groudon and a kyogre
Sorry, Groudon is specific to HeartGold but is you can get a HeartGold user (MAYBE ME) to lend you their groudon, you can get a Rayquaza! Rayquaza's better anyways
You can only get a rayquaza by transferring it from heartgold or soulsilver or by the GTS.
shiny rayquaza with wondergurad
Please see the related link below. Catching Suicine in SoulSilver is the same as catching it in HeartGold.
nothing
Groudon isn't in HeartGold, it is in SoulSilver. Kyogre is in HeartGold as well as Rayquaza.
Rayquaza can be taught HM02-Fly. In HeartGold and SoulSilver, Rayquaza learns Fly at level 45. In Black and White, Rayquaza learns Fly at level 65.
You can't without hacking, unless you have HeartGold and SoulSilver. Why, one word, HM Moves, Rayquaza can't migrate, you need to get Rayquaza on the Nintendo DS. (This requires HeartGold and SoulSilver. You need both, you'll see why...)