1. It all depends... Some places do have competitions for Pokémon, usually at shopping malls or stores like Toys R Us. You would have to look around at some of the places in your area and see if they have any competitions Maybe even video game stores like GameStop would have some info. Here in Northeastern Ohio, (where I live) Pokémon is not as popular as it once was, and competitions are pretty rare, the Toys R Us by my house had acouple or tournaments right after Fire Red/Leaf Green was released, but that has been about it.
2. The Elite 4 in pretty much still the quickest way to level up. However, if you want to seriously train your Pokémon, and you are not just raising your Pokémon to complete your Pokédex, you will want to EV train your Pokémon before you start battling the Elite 4, so that you won't mess up your Pokémon's stat distribution.
3. No, shinies used to have certain set stats back in the GSC days. (they had above average stats, but they didn't have max stats.) Also, breeding a shiny Pokémon with a non-shiny would give you a 1 in 64 chance of getting a shiny baby. Now, in the Advanced Generation games, shiny Pokémon do not have any set stat values, they can have any stats/personalities/genders/etc.... Also, breeding shiny Pokémon will not increase your chances of getting a shiny baby. Getting a shiny is a totally random occurance, there is about a 1 in 8192 chance of finding a shiny, no matter what you do. (Catching in the wild, hatching from an egg, getting a Pokémon from somebody, such as the Beldum you get from Steven after you defeat him and the Elite 4, etc...) So basically, shiny Pokémon are nothing more than "trophy" Pokémon. They may have good personalities and good stats, or they may not. There really is no bonus for getting a shiny, outside of the fact that they are nice to look at!!
4. Level 40 Pokémon should be able to get through the Elite 4, although you will want to keep some healing items and revives with you. Basically, you can beat the game with almost any Pokémon team, you can use your favorites, just try to balance out your types. Water/Ice Pokémon will help you a lot in beating the Elite 4, as well as Electric and Fire types.