I feel like this is almost impossible to answer. It's just too broad really.
Let's suppose that the colony land on something similar to a desert (sand desert or cold desert, whatever)
Ok. Well if it's a "sand desert", presumably comparable to the "sand deserts" on Earth, it's really, really hot and really, really dry. So they would need much more water than I would expect other colonies would need, and they would require something to keep them from overheating. That could be anything from housing with central A/C to cool clothing.
And if it's a "cold desert", presumably comparable to the "cold deserts" on Earth, it's really, really cold. So the opposite would apply -- they would need something to keep them warm, from central heating to layers of clothing.
I'm assuming they get clothes, but I'll get to that later.
with a limited amount of natural resources. I assume they have at least access to unlimited breathable air and clean water source.
In a "sand desert", they very well may not have access to an unlimited clean water source. Either way, "natural resources" can vary too widely to really predict what they would need. For example, assuming that they have sand and water, they could theoretically build sand-castle shelters, or more likely something adobe-esque. They probably even have access to real stone. But if it's a "cold desert" like Antarctica, they have to build something more like igloos instead. Same concept, but completely different resources. Lots of things can be made from earth or earthen materials that can't be made of ice, as ice tends to have a relatively low hardness on Moh's scale. It also tends to melt when you hold it, so most tools wouldn't be able to have ice handles, at least not for very long.
They also start with the current human knowledge and enough supply for the first generation of humans (but no seeds and no animals).
Enough supply of what? Food? Does "enough supply" also cover things like clothing, medicine, tools of any sort? Having seeds (and perhaps animals too) seems irrelevant, since most plants (and perhaps animals) probably wouldn't be able to survive in either a sand or a cold desert, given the lack of arable soil (and food/water for the animals).
In order to survive (i. e. avoid extinction in the next decades/centuries), what is needed in the fauna and flora of this desert?
The plants and animals of this desert would have to be things that could be easily domesticated, meaning that the humans would have to be able to understand them relatively quickly and control them well enough to control their population. Oh, and they have to be edible -- can't have things that kill us when we eat them.
Beyond that, the plants and animals of the desert would have to provide the group with any other materials that they can't get directly from the earth -- fibers and leathers for clothing as well as things like sacks, something sturdier (like wood, though that's unlikely in this setting, so probably bone) for use as tools, and kindling for a fire (if not for warmth, then for cooking).
Can they survive without any birds, fish or mammals (reptiles or insects only for example)?
In a desert, where I'm assuming trees are scarce, maybe. Without wood or bones (or stone in an Antarctica setting), you'd have virtually no tools. This also removes leather as an option for clothing (edit: unless you kill a lot of snakes), which leaves you relying solely on fibers of plants, which you would have no easy way of weaving together.
How many differents kind of vegetables and fruits do they need?
I can't really address this one, mostly because I don't even begin to understand the way ecosystems can work. As far as the human diet goes, I would assume they would only need one edible plant of which they could control the reproduction.
They don't have to live a healthy life, just enough to reproduce, raise new children (even with a high infant mortality) and perpetuate the human race.
While this has nothing to do with the minimum requirements of the local environment, I feel like the number of people in your initial colony would be immensely important.
If you started with just two people, for example, the chances of birth defects in future generations would be pretty high considering the tiny size of the gene pool. A colony like this may not be able to survive just because of that alone.
However, if you have many people, say 10, then you end up having to account for social aspects of the colony. Mostly there's the issue of who's worth keeping around and who isn't. Who is expendable, not only because they waste resources, but also because they can provide resources -- humans are animals too, after all, and animals can provide many things like bone, food, and leather. I would actually be willing to bet that you wouldn't need much of a local environment if your colony were willing to divide into two -- the humans and the animals.