What is the minimum number of colors needed to sticker a Futtminx such that:

1) The same color is never on two or more sides of one piece
2) The solution is unique (i.e. there are no duplicated pieces)

I don't even know where to begin...

There are 90 edges. Since they can not be flipped in place, it is acceptable to have mirrored pairs. So we need to find 45 unique combinations of 2 colors, without any colors being picked twice. This means that you need at least 10 colors to identify the edges (10 nCr 2 = 45).
For the corners you'd need at least 7 colors (corners may be mirrored, so you need 30 unique pairs of 3 colors. 7 nCr 3 = 35, 6 nCr 3= 20).
The orientation of the core can surely be determined by those 10 colors, since there are 12*6=72 ways of orienting the core, but far more ways to place the stickers.

This does not guarantee that it can be done using 10 colors, it just shows that 10 colors is the lower limit for your question. It shows that using 10 colors you can uniquely identify every piece on a Tuttminx, but it does not guarantee that that Tuttminx will ever be solvable.

Edges can't be flipped on a regular Tuttminx, but they can on a Futtminx!

Tuttminx has two types of edges: pent-hex and hex-hex. The former cannot be flipped, the latter can.

Quite. The Tuttminx is like a Dino Dodecahedron if you forget about the pentagonal faces. Therefore the edges can be flipped.