no
There is not a weather institute in Pokemon Heartgold, you are thinking of Pokemon ruby, sapphire, and emerald, there is not a weather institute in Pokemon heartgold.
You beat the Pokemon league and then on the tv it appears that a flying Pokemon appears which is Rayquazza and its level 70 and then you find it on top of the Sky Tower and catch it
The Weather Institute is located to the left of Fortree City.
The Weather Institute is not in Heart Gold. It is in Ruby, Sapphire, and Emerald.
beat the grunts
On the Route soutwest from Fortree city.
It is not a selectable Pokémon in Ruby. If in doubt as to which characters are "4th generation" you can consult the Pokédex website. Gible didn't exist in the original Ruby game. In Pokemon Omega Ruby, however, you can catch a Gible (as a Hidden Pokemon) on Route 111.
Route 208 and 210, but only appears if you have Pokemon Ruby in your GBA slot.
Celio is not a character that appears in Pokémon Ruby instead he is a character that appears in Pokémon FireRed and Pokémon LeafGreen he appears in the Pokémon Center on One Island and is in charge of the Network Machine you as the Player Character are required to find the Ruby and Sapphire items in order for him to finish work on the Network Machine so that your games can trade with games such as Ruby, Sapphire and Emerald.
You can capture Feebas on Route 119 in Pokemon Ruby, this is dependant on the phrase in Dewford (that changes it's location) and is luck to find the correct tile that Feebas appears on when fishing.
it is left of fortree city
Pokemon ruby's version IS Pokemon ruby.