Yes, you can choose to. Why would you though. You are merely pretending that your warrior cannot lockpick or use magic - but no matter how hard you pretend, he CAN use pretty much everything. So why would he/she, as a thinking person, not use them?
Why would your thief not use daedric stuff when he can use them just as fine as leather? Why not use restoration while he KNOWS how to cast restoration spells?
Can you explain that? How does it make sensse, apart from your character being intentionally blind to his/her abilities or having phobias about magic and lockpicks and stuff?
Again, this is a sandbox RPG. The word sandbox means that the system does not limit yourself neither the char. It just gives you a set of skills, and a world with some things going on (politics, dragonborn, secondary quests...).
In an online game, the game MUST limit stuff and make sure it's all balanced. That's NOT needed on a game designed to play by yourself. If Skyrim was online it would be a complete disaster, simply because everyone would end up having ALL skills and perks, to take all the possible advantage.
So, they would HAVE to set a skill limit and perk limit, and balance skills accordingly.
Have you ever played a mmorpg? If I want to play a warrior, I have to focus on warrior skills if I want it to be a good warrior. If I decide to use some other skill, it will mean I'm lacking something on the warrior side (my choice anyway), and if I decide to use some other skill I will probably have to sacrifice some other warrior skills. This is what balances a game, but this is only needed on online games, where lot of players are going to play together and the system needs to control what the player does. To say it another way, the system enforces the rules.
That's not needed in Skyrim.
Skyrim can't "know" that you are playing a heavy warrior that can't lockpick or sneak. You will be able to use the ability and the game will increase the skill as you do. In an online game this would mean you have less points to invest in your warrior skills so you probably wouldn't use those abilities.
You can do the same in Skyrim, instead of "abusing" the sandbox mode system and complain because your character is powerful.
You mention the use of lockpicking and restoration spells. Well, let me put as example the character I play.
I wanted a sneaky archer (I usually start these games as a melee warrior, so I wanted a change). I wanted it to be good-aligned (not evil), with knowledge on locks and merchandise (speechcraft). I though on making it light-armored.
So, with this choice, I always use bow. I only wear light armour or clothes. I lockpick chests and doors on caves and ruins. I sneak here and there to try surprise my enemies. I never read a single spell book (I didn't choose magic for this char), neither craft a single potion, weapon or armour. Also, I never pickpocket anyone neither steal items, even if they are flagged white. If it's not mine, I don't take it.
I only pick perks from sneaking, archery, lockpicking and merchandise. Light armour barely increased because I don't get hit a lot, but I pick perks there too when I can.
When I create the [censored] warrior I have planed, I won't have problem stealing from shelves if noone is looking, or asking for a reward without shame. I will wear all those armours I can't wear now, and use different weapons I don't use now. I won't use lockpicking neither sneaking, so I will have to approach combat and some situations another way.
It's funny to read comments like: "how am I supposed to not gain sneaking?"... The common answer is "don't use it". The thing also is... why you use it? Because you know you're going to sneak past someone. If you don't want sneaking because it feels powerful or doesn't fit your char... then why you want to sneak past someone?.
Also if you learn a healing spell and use it to heal, even if you're a warrior, you will end up gaining a lot of restoration.. and if you pick perks, you will end up complaining that game is too easy because you can heal with no problems and never die. The question here is not to ask why the game allows that, but why you use it with that character profile.