What wouldo you want to do? What would you want to do?

What wouldo you want to do? What would you want to do?

What do you want towould you do? What would you do?

You're restating the original question, "What would you do (if you were in visible)?"

What would you do if you were invisible? What would you do if you were invisible?

What would you do if you were invisible? What would you do if you were invisible?

What would you do if you were invisible? What would you do if you were invisible?

I would probably sneak to the backyard of my favorite restaurant and research their recipes. I would probably sneak to the backyard of my favorite restaurant and research their recipes.

I would probably sneak out to the backyard of my favorite restaurant and research a recipe. I would probably sneak out to the backyard of my favorite restaurant and research a recipe.

I would probably sneak into the backyard of my favorite restaurant and research their recipe. I would probably sneak into the backyard of my favorite restaurant and research their recipe.

Backyard, an object, needs an article. Recipe, another object, also needs an article. You can just use "the" again, but I used "their," which is better because it's more clear. If you want words in place of 'research', you could say "steal," "copy," "memorise,"

I would probably sneak into the backyard of my favorite restaurant and research their recipe. I would probably sneak into the backyard of my favorite restaurant and research their recipe.

I would use "sneak into the..." as it has the feeling of infiltration. Like a ninja. "Backyard" gets the point across but I think just "back" or "back alley" of the restaurant sounds more natural. Backyard has more of a "lawn, swimming pool, behind-the-house" feel to it.

Ratatouille is my favorite food in particular at the restaurant, but I can't cook like thate restaurants can. Ratatouille is my favorite food in particular, but I can't cook like the restaurants can.

Ratatouille is my favorite food indish, particularly at theis restaurant but I can't it cook like that. Ratatouille is my favorite dish, particularly at this restaurant but I can't it cook like that.

Ratatouille, in particular, is my favorite food in particular at theat that restaurant, but I can't cook like thatem. Ratatouille, in particular, is my favorite food at that restaurant, but I can't cook like them.

Because you've already introduced the restaurant, it makes more sense to say THAT restaurant, instead of just the restaurant. I can't cook like that is okay, but them is better because it's clearer.

Ratatouille, particularly, is my favorite food in particulardish at theat restaurant but I can't cook like thatey can. Ratatouille, particularly, is my favorite dish at that restaurant but I can't cook like they can.

I would use "dish" over "food" when you're talking about a single prepared menu item. And this sentence structure has a more natural flow in my opinion.