Do gemstones have a purpose in Skyrim?

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.

I found another use for gemstones (could possibly be considered a spoiler):

The Atronach Forge in The Midden under Winterhold College can be used to conjure atronachs. You need to put "recipe" items in the offering box and then pull the lever. One of the recipe items is a gemstone.

The "Atronach Manual" lists the following recipes:

Flame Atronach: Fire Salts + Ruby
Frost Atronach: Frost Salts + Sapphire
Storm Atronach: Void Salts + Amethyst

Also, gemstones + soul gem + salt pile can make either frost salt, fire salt or void salts, depending on the gem.

Yes. With the smithing skill you can create gemmed jewelry like rings and necklaces that you can enchant that are useful for mages.