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Creating a Mii Fighter in Smash Ultimate
1. Build yourself a Mii
Navigate to System Settings
Go down to 'Mii' and select 'Create/Edit a Mii'
I'd suggest using 'Choose a Look-Alike'
Press 'Y' to shuffle the random faces until you find a good template
Tweak him until he looks just right, give him a name and you're good to go!
2. Build yourself a Mii Fighter
Pop open the game (in your boyfriend's account) and navigate to Games & More
Select "Mii Fighter"
Choose a fighter type, moveset, etc. and then customize to your heart's content!
Some answers to your questions...
I heard that in order to even use the Mii fighter characters, you need to unlock them through something called "story-mode"?
Unlocking the different Mii Fighters through World of Light (WoL), is one way to have fighters be created for you (and the only way to earn the "default" Mii Fighters). But WoL is not needed to create Mii Fighters. You can follow the above steps without ever playing WoL.
I want to create a Mii character that looks like him, and heard that you can create one from a photo? I found something called "Mii Maker" that says you can do this by taking a picture with the Wii U GamePad camera - however no one I know owns a Wii..... is there another way to make a Mii fighter character from a photo without using the Wii console?
Minor distinction, but the Wii and Wii U were actually two separate consoles. So you wouldn't be looking for someone with a Wii, but the even more obscure Wii U! (this is assuming there's a way to even transfer Mii's over, which I haven't looked into)
Instead of going that route, I'd use the "Choose a Look-Alike" option I detailed above. Keep shuffling until you find a generated Mii that looks close enough, and then go in to tweak whatever needs fixing.
I would like this to be a surprise - so if I am able to create a character - is it possible to create it in another console, or in another account, and then transfer it to him somehow, on his birthday? QR code? Send as a gift?
I've read that this can be done using Amiibo--you would create the Mii Fighter on one Switch, save it to the Amiibo, and then use that to transfer it to another. But I haven't tested it, so I can't guarantee it's possible.
The biggest problem you'll run into with this secrecy is that Mii Fighters only show on the character selection screen if you've created one of that type. So, for example, if you create a Mii Swordfighter when before he had no Mii Fighters on his system, the selection screen may look like this:
Notice how there's just a Mii Swordfighter there at the end. If he's used to seeing the selection screen without, then this may be a dead giveaway. So be careful!
There seem to be numerous varying opinions on which Mii fighter is best: Gunner, Swordfighter, or Brawler. My bf mains Bayonetta, and will sometimes use Dark Pit or Dr. Mario as alternatives - so which do you think would be the best choice for someone who uses these characters?
If he mains Bayonetta, Dark Pit and Dr. Mario, it sounds like your boyfriend is a brawler. All of those characters want to be up close and personal so they can combo the crap out of you. They all only have one real projectile, and that's only used to let them get close, so Gunner's out. None of them have swords or disjoints for carefully spacing approaches/attacks, so Swordfighter's out.
Therefore, I'd suggest Mii Brawler as the best fit for your boyfriend from what you've told us. Brawler (like the name implies), does best when he's right up in your face. Some of his movesets also include a single projectile to help him get close, just like your boyfriend's characters.
Good luck!!