As @Willk suggested (blessed be), i will extend a bit on my comment.
What you described is a common phenomenon when an humid air mass from the sea encounters a natural barrier like a mountain range. The air rises, is unable to go through and falls in one side as heavy rains.
At the other side of the mountain range, zero or reduced rain ever falls. Forming a Rain shadow.
In places near the tropics, the high amount of rain in one side contributes to the formation of heavily forested areas, while the other side is prone to desertification.
The picture above is of the Tibetan Plateau, with the Tarim Basin being the deserted area, a subtropical forest area below and the Himalayas between them. This is an example of a combo for a colder climate like you described. Both the forest and the desert area are colder!
For another example, we have the division between the Amazon Rainforest and the Atacama Desert:
At one side we have the famously hot and humid Amazon Rainforest on Bolivia, while at the other there is the Atacama Desert, perhaps the most arid place on earth while being cold! The secret here is that the coastal Chilean mountains blocks air from the ocean also, creating a Temperature Inversion making the desert frigid.
In essence: Yes to both questions!
Here is a link to an exceptional article from Wikipedia detailing rain shadow.
Edit: As requested, examples of hot deserts.
This is the Chihuahuan Desert (and surroundings), a hot region surrounded by stretches of forested area. In fact most desertic areas of North America are pretty hot and suffer from the effect of rain shadow from the Rocky Mountains. While I suspect the greens around are not rainforests per se, they could be in your world, no problems!