Gamertags with a brand name, and publicly showing my gamertag to the world

it has other meanings too

The word 'Pepsi' comes from the root word dyspepsia, which means indigestion

you should be ok lol.

When you change the gamertag. Everything shows the new gamertag expect the name in the xbox live party. It says my old gamertag

Not every gamertag will have a publicly viewable Xbox profile. It could be old accounts from Zune, GFWL, or etc.

The best way to check for a gamertag is to input one on the Xbox account signup page, or the change-gamertag page. Third party websites like "checkgamertag.com" don't have full (or even actual) access to the gamertag database.

If it's not available, it's not available. You'll have to choose another name. Gamertags are only released if the owner has not logged into their Microsoft account in five years.

When I log in to Bungie, it shows my old gamertag that I signed up with, as my Bungie account name, aswell as all of my fileshare, the author is my old gamertag, which is quite annoying. But it has my gamertag next to the account name.

Did you change the gamertag or your account name? The name that it shows below the gamertag on the console is your name.