A cyan pony, a purple unicorn, a black pegasus, a green unicorn and a pink dragon are all necessary, though in what numbers are by your choice.
While this request may sound ludicrous, most of it is not, depending on how flexible we and Mrs. Mars is willing to be with the definition of dragon, unicorn... and especially pegasus. In fact, we do not need much in the way of genengineering to get the first two, even in their desired colors.
That pegasus though, it's going to require a major redesign. In fact, I strongly suggest that everything else be real and placed as living animals within this 'paradise', however that the pegasus be restricted to non-living technology. Holograms for viewing black pegasi in flight; animatronics for black pegasi which can be touched and seen up close.
Here are possibly ideal foundation species to shape into the desired creatures -
Pink dragons:
21st century Earth had naturally occuring large pink iguanas. Some, like the marine iguana, were only partially pink in males during the breeding season.
http://www.arkive.org/galapagos-marine-iguana/amblyrhynchus-cristatus/image-G8114.htmlOthers, like the pink iguana, were pink year round for both genders. And conveniently didn't need to spend much of their lives underwater.
http://www.treehugger.com/natural-sciences/this-sneaky-pink-iguana-evaded-charles-darwins-attention.htmlAny desired increase in draconic traits can be worked on from one of these bases.
unicorns:
If a spiral horn is desired, blackbuck.
http://en.wikipedia.org/wiki/BlackbuckIf a straight horn is desired, Oryx.
http://www.awf.org/wildlife-conservation/oryxAny horned species could be manipulated.
http://www.youtube.com/watch?v=TBesVO_3iQkA surgical answer would be the simplest. A horned species, possibly altered through breeding or more direct genetic alteration to make it more equine and less antelope-like or goat-like, would soon after birth have the horn buds removed. One horn could be turned (so the horn grows with a backwards curve, rather than a sideways curve natural to most species) and grafted to the center of the animal's head rather than the sides where it had been harvested from. After a fairly short time for healing, the creature would appear to be a natural single-horned animal.
http://www.youtube.com/watch?v=TBesVO_3iQkIf this is absolutely unacceptable, then some work could be done to determine what genetic controls exist for the placement and spacing of the cells fated to become horns. With rather a lot of trial and error, these controls can be identified and manipulated to produce 'natural', from birth single horns with the desired curve, spiralness, and shape. The problem is the generation times. This sort of trial and error work is best done with bacteria, with twenty minute generations; to make this solution workable in horses, goats, or antelope would require probable hundreds to thousands of their generations for perfect control.
Teal, purple, and green furred mammals
Color is actually less challenging than the placement and number of horns, as animal pigments exist in all the requested colors, especially in birds.
For teal, perhaps the racket-tailed roller
http://www.tanzaniabirds.net/African_birds/roller_racket-tailed/rtr.htmFor purple, perhaps one of the species of lamprotornis, maybe the 'purple starling'.
http://en.wikipedia.org/wiki/Lamprotornis For green, perhaps the lesser green broadbill
http://www.lpzoo.org/animals/factsheet/lesser-green-broadbillThe genes controling the pigment would need to be isolated, and then placed with specific promotors for hair formation already existing within the goat, antelope, or horse species used to form our 'unicorn' stock to allow the proteins that make the colors to appear in the fur. This is a natural process we would 'borrow' to make the fur appear as we wish; we would 'knock out' the existing pigments that might mask the color we want to show, and we would 'knock in' the sequences for the colors we wanted instead.