yes
You can only get (one time) access to the Rainbow Fountain through Fountain Faerie Quests, and these quests are random events, just like any other Uber Faerie Quest. See link for more details.
The rainbow fountain only works if you have completed a fountain faerie quest. If you have completed a fountain faerie quest, go to the rainbow fountain and you can paint your neopet any color except MSP, Royal, and Robot. Bori cannot be painted ice because the ice bori was a special "Hannah and the Ice Caves" release.
There used to be a time when you would sign up for sponsors which would give you Fountain Passes. There was <i>also</i> a rare quest from the Fountain Faerie to get her very expensive items in return for a Fountain Pass. These Fountain Passes allowed you to get a paintbrush of any kind from the Fountain Faerie. However, the fountain doesn't work anymore, probably because people could just buy the most expensive brush and make billions of neopoints quickly, throwing off the balance of the game.
Any type of random event that alters your pet(ex.boochi random event,fountain faerie quest,etc.) is VERY rare.
Any faerie quests on Neopets are completely random. You must wait for them to appear.
It only works when you complete a quest from the fountain faerie. After you do, you can visit the fountain and paint your pet any color you'd like. :)
Spin lots of wheels and complete Faerie quests. If ur tired of waiting for a faerie quest, just go to Meridell or Faerie Island and there are a couple faeries there.
I use a side account to search and buy the item (for any Faerie quest) using the Shopwizard, then I give/trade it to my main account.
Basically, you just have to wait for a random event if you want a quest from the Air, Battle, Crafting, Dark, Earth, Fire, Light, Water, Space, Fountain, or Queen faeries unless you want to spend neocash. If you are willing to spend Neocash, you can buy a Faerie Quest Fortune Cookie when they are available. They guarantee you one faerie quest a day from any one of the above faeries (except the Crafting Faerie) for a week. It is also worth noting that there are three other special faeries: the Snow Faerie, Jhudora, and Illusen. The Snow Faerie will grant you a quest whenever you visit her on top of Terror Mountain. Jhudora grants a quest every 12 hours if you visit her on her Bluff in Faerieland, so long as you also have not done a quest for Illusen in that time. Likewise, lllusen grants a quest every 12 hours from her Glade in Meridell, as long as you haven't done a quest for Jhudora in that time.
http://www.youtube.com/watch?v=k_82RWJgmgM
The same thing any other neopet would, it just sits on your account really, until you interact with him/her.
New neopets are created periodically so it is entirely possible, but unknown at the time.