You can absolutely wear a trench coat during the winter, especially if you live somewhere with mild winters! It is not a great idea to wear a trench coat alone as your only protection from the cold though. Wear layers or a thick sweater and your trench coat should then suit you well in the winter.
Honestly, you can wear a trench coat during the summer, fall, winter, or spring depending on where you live and the material it is made out of. For colder weather, winter or wool are the best materials to keep you warm, and for warmer weather, cotton or polyester are the better choices.
Yes, trench coats are still very much in style today. They are timeless. A lot of models and well-known public figures wear trench coats on a regular basis. Trench coats of a wide range of colors are currently in style, but the most commonly worn colors are beige and the traditional khaki.
A trench coat is meant to be worn as the outside layer of clothing just like an overcoat, except a trench coat is more suitable for rain rather than overcoats which are made to withstand cold weather.
Overcoats are meant to be worn as the outermost layer of your clothing. They are supposed to be comfortable, easy to maintain, and attractive. Keep in mind that overcoats are longer than an average coat and should be longer than any layers that are underneath them.