I think it would be nice to make language skills useful rather than removing them.
Unfortunately, there is no viable path to make them useful. Even if we add a ton of quests for each language skill, they would still suck for many reasons. They are just too specific and there are not enough creatures of these types to be of any use. Best way to handle languages can be through Perks in my opinion, if we ever implement them. And perhaps some skill combinations can work as Jay_H proposed.
Alchemy could be implemented as a character skill.
Certainly! And quite close to what I was thinking! Each Potion will have a few levels, so to speak. Depending on your skill, you'll be able to create a certain grade potion. I am also thinking that in the future poisons/weapon oils can be added, as well as bombs potentially, in the same Alchemy skill. Potionmakers levels can vary depending on the quality of shop, as well as their prices.
The Critical Strike skill could be eliminated and its effects implemented in the rise of each individual weapon skill.
Yeah, critical strike can be part of Weapon Skill + Luck, I don't see a real need for a skill like that.
Running, Jumping, and Climbing could all be put together into one;
I am actually thinking of Athletics for Running and Swimming, as well as Acrobatics for Jumping, Climbing and potentially Dodging, though I personally don't like dice rolls at all

But it should be an option. These skills can go well along with Stamina meter, where instead of increasing Speed, they would actually reduce Stamina consumption instead, while Speed will be handled with the Attribute. Acrobatics should still increase jump height, though.
Streetwise and Etiquette
Well, they can be either kept or replaced with Speech and once again, Perks might grant even more variety in what direction you take your character. Perhaps, even Mercantile can be merged into one to create a Super Social skill, while different perks will allow you to take character in different directions. But maybe not, there is at least some value to these skills, even if it's not huge.
All melee weapon skills could be merged into either One-Handed or Two-Handed types, similar to TESV. Hand-to-Hand and Archery would retain their own skills.
Yep, that's good idea and what I was thinking. Perhaps, Polearms might be added at some point. Archery can renamed to Marksmanship and include other weapons as well - Crossbows, Throwing Knifes, etc.
A new skill could be created for armor, an analogue to Dodging. After all damage calculations are made, a net reduction could be applied based on Armor skill when an armored slot of the character is attacked, with trigger chance and damage reduction improving as skill rises.
This is probably one thing I am opposed to, and I thought about this for awhile. At first, I thought it was a good idea, but after consideration, it makes no sense that Leather armor would protect you better than Plate, even if you are highly skilled. And that's how it worked in later TES games. There is not much use for Armor skills, at least I don't really see them.
Armor should certainly affect your Speed and style of gameplay, perhaps add some Perks for it as well, but my idea is to tie Armor to Equipment Weight and it will determine how quickly you move in it. Equipment weight will rise with Strength, so high Strength characters will move faster in heavier armors. And a few specialization Perks can take away any real need for Armor skills.
Pickpocketing and lockpicking could certainly be merged, and perhaps Stealth as well.
Yep, especially if we add locked chests to dungeons, it might be a good idea. Warriors will break locks, Mages unlock them with spells, thiefs will use their skill. Potentially traps and possibility to disarm them as well. These three can be either two skills or even just one.
Enchanting could be an additional skill, like in TES3-5.
Most certainly, and there can be many interesting mechanics with it. Perhaps you'll need to break enchanted items to learn enchantments, but you'll not need to identify items which enchantments you know already.
Maybe a Survival skill, which would save the character from life-threatening strikes such as poisons, deadly diseases, and critical strikes when at low health. It wouldn't trigger all the time, but would increase with skill.
Survival is an interesting idea and can be used in variety of ways. It can determine rate of random encounters when travelling recklessly, speed of regeneration when camping, maybe even some other things, like cooking. A lot can go into this one!
If we implement spell tomes, Necromancy could be a new spell school. No one would sell their spells, but you could occasionally find spell tomes that teach them to you. Raise Skeleton, Bind Ghost, Pacify Undead, Invisible to Undead, and so on.
We can probably rebalance existing schools, without creating new one, but I certainly wouldn't mind the new spells and magical specializations in the future.
Great suggestions overall, thank you, Jay_H!