You could use either term interchangeably. I would start the sentence by saying, "In winter, taking out the trash can be a hassle." However, like I said, you can use either term. As a native English speaker, I tend to say "in winter" versus "in the winter."
The sentence, "Taking out the trash can be a hassle in the winter/in winter.", is not how a native speaker would usually say this. This would typically be said as: "Taking out the trash can be a hassle in the wintertime."
But if you went ahead and said it as: "Taking out the trash can be a hassle in the winter.", you might get a funny look from a native speaker. This sounds a little
They are both correct in your sentence. You have to say in the winter if the reference is to a specific winter: What did he do in the winter? (= last winter)