You need something like an integration sphere.  Point several LEDs into it, use collimating lens to couple the outgoing light into fiber. But integrating spheres are expensive. Instead you may want to use some diffuser and the same principle - shine on it with many LEDs and collimate diffused light. Or use homogenizer rod, which works on total internal reflection. What solution maybe doable for you depends in first place on the fiber (SM/MM) and desired optical power. Due to their low brightness LEDs are difficult to couple to fiber.

You want to overlap the foci of the LED on the fiber end. That's a problem for even one LED as they're incoherent, so the best you can do is to image the emitter on the fiber face. That's still a problem if you have an LED with a wire bond covering part of the emitter. How big is the fiber? Is the LED an edge or top emitter?

Since the wavelengths are different (you don't say what they are, but let's assume RGB) you can use a prism to get them to overlap spatially, then focus them into the fiber. Expect losses.

No idea what you mean by mixing. LED's aren't usually intense enough to cause any non-linearities.

Few manufacturers provide standard multi wavelength LEDs. http://www.epitex.com/products/stem/stem_index.php

Couple  such an LED directly to a large core fiber (plastic fiber) . Use a collimator (plastic lenses) lens at the output side   to focus it to a MMF of smaller diameter or to the area which to be illuminated.

Normally, the LEDs have low power, which is a main problem to couple its light into the fibre. You can use a collimator with large core fibre, such as 1mm polymer optical fibre.