People from deserts did plenty of conquering in history. Infact in the actual middle ages, most of the big conquering took place at the hands of desert or semi-desert dwelling Arabs, Turks, and Mongols.
Examples here. Turbaned helmets provide head-protection against sun and sand as well as arrows and blows. Metal armors are heavy and get hot in the sun, many nomad armies wore quilted, leather, or silk armors.
The bow is the weapon of the nomadic horse-archer. Living in near-desert conditions means living with livestock, not farming. Living with livestock means finding some livestock to ride. When your whole army is riding, you can use mobility to your advantage. Move, shoot, move, don't close with the enemy unless their morale has already been shattered.
Camels, obviously, are great in deserts. But the horse family is actually pretty well desert adapted as well. One of the most famous horse breed is the Arabian, and its quite a stretch to call Arabia a grassland. The Arabian horse is ideally situated for the nomad warrior, having great stamina and not being too big (i.e. not needing to eat and drink alot like the destriers of the West). Plenty of horse-like animals live primarily in the desert like the Khulan or Imperial Zebra.
Going more fantastically, creatures often imagined to live in deserts would include giant lizards, griffins, hippogriffs, giant scarabs, and the phoenix.
Outside the Desert
They would fight...the same! As long as there are open plains, the horse-arching tactics of the nomads will work. The Huns, Avars, and Magyars penetrated as far east as Hungary while raiding into France, the Arabs and Berbers conquered Spain while raiding into..France. The Mughals (Mostly Afghans and Turks) conquered North India and raided into Southern India. The Mongols conquered northern China while still primarily nomadic (by Kublai Khan's day their armies were Sinified), and raided Korea and southern China. While dense forest regions like South India, South China, and European Russia remained unconquered, they were all certainly subjected to slave raiding and tribute. Forest wasn't much of a barrier to well organized nomad armies.
Why would your desert people have them? They live off livestock and livestock gots to move. No time to settle down for cities. A few walled trading posts of a few thousand on strategic trade routes is all that the desert kingdom will get.