You get all the possible cooking recipes from the start: it just depends on what ingredients you have on hand. When you visit a cooking pot, you'll see all the available recipes, and when you have the right ingredients, the recipes will enable.
The stews and soup that you can make with cooking sell for a few gold and have some basic perks, but are not nearly as good as potions. In terms of what you can make with carrots and potatoes, there's:
Beef stew
carrots, garlic, raw beef, and salt pile
Increases max stamina by 25, stamina regeneration for 12 minutes
Cabbage potato stew
cabbage, leeks, potatoes, and salt pile
Restores 10 health and stamina
Vegetable soup
cabbage, leeks, potatoes, and tomatoes
Health and stamina regeneration for 12 minutes
Venison stew
leeks, potatoes, salt pile, and venison
Restores 15 stamina, health and stamina regeneration for 12 minutes
If you scroll over the other recipes while using a cooking pot, the items you need to make them are listed under the item detail box.
The horse can't "disappear" permanently, just like your companions can't. If you somehow lose track of your horse, you simply need to fast travel somewhere, and the horse will show up there, too.
If you fast travel inside a city, the horse will usually be teleported to a stable outside the city or to some other location around there if the city doesn't have one.
However, horses in this game are particularly proud (they're nord horses after all), so what might happen is that your horse runs away, gets attacked by something and tries to defend itself, dying in the process. Sadly, the "fast travel" trick won't work if the horse is dead.
There is also a random stray horse, free to take without stealing in between Valtheim Towers and Ivarstead
Best Answer
Gems are useful for smithing, as they increase the skill by quite a bit. They can be smithed into jewelry which can then be enchanted with properties that buff certain attributes (fortify light armor, extra sneaking or lockpicking percentage, etc).
Also, some gems are used during quests. For example, there is an Argonian in Riften who will ask you to find three Flawless Amethysts so that he can make a wedding ring for his lover.
As you mentioned, they can also be sold for a fair amount of gold. Keep in mind though that gemmed jewelry is more costly than just gems alone. If it's money you're after crafting jewelry and selling that would be more advantageous than selling the gems alone.