Is this jacket waterproof?

No. It is a cotton blend shell with no membrane or sealed seams, so it handles light drizzle at best. It is not a rain jacket.

No. It has a stand-up collar for neck coverage, so you would need a separate hat or a hooded layer underneath in wet or cold weather.

The cut is loose with long sleeves, so it runs roomy and leaves space for a sweatshirt or hoodie underneath. The Velcro cuffs let you tighten the sleeves.

What is the badge on the chest?

It is an embroidered Lamar County Sheriff's Office patch on the front chest. It is decorative and part of the military styling.

A zip runs up the front and is covered by a Velcro placket. The cuffs also fasten with Velcro straps.

Is it good for cold weather?

It works as a mid-layer in cool weather but has no insulation or windproofing. In real cold you would want a fleece or insulated jacket under or over it.