Constructing a desert area

I'm trying to construct a most realistic desert possible. I've made a continent with my little knowledge and may ask further questions about climate and similar stuff, but now I want to center on a desert area that I want to locate.

Our little world

I have a large continent, large as Europe and Asia together.

It has a mountain change that divides the continent almost in half, except for a few spots where there is lower terrain and in the northernmost part of the continent the mountain chain dies. I want there to be a forest on both west and east sides of the mountains, being more vast in the southwestern part.

I want to focus on the west side, having a great green land on the south with a template climate and as you approach the mountains colder temperatures would be found, and in the north side a more cold green land that advance into a boreal area.

I want to somehow get a desert between them; the desert may occupy from the west coast to the mountains on the east part (like Sahara in Africa).

Questions

1. Is it possible given the actual conditions I've just given to you?
2. Would it be possible for the desert to have one or two large rivers?
3. Where can I find large amounts of trees on a desert like this?
4. Where may I find large mines?

(*I have little knowledge about how large amounts of minerals are created so this question may be a stupid one or minerals would depend on factors other than just position)

Edit:

Here's a map of how I imagine my world.

Gretland is a vastly green land, has lot of agriculture zones, and template climate. Imagine Gretland climate as southamerica-like.

Winthold is more like a Canada climate place.

Little triangles are mountains.

I want my desert to be in Thaliak.

More info will be provided if needed