We have a 16' Collin-Arndt gooseneck stock trailer we use to do all the things you mention. Collin-Arndt is a local manufacturer who has a nationwide dealer network. We bought direct since they are 15 miles away. Steel frame, all aluminum upper structure, PT floor covered with stall mats and lined (safer for the horses and the aluminum walls) up to about shoulder height. Above that it's open. We have plastic windows that install from the outside with a few screws for use in winter. The gooseneck part of the trailer holds a plastic water container shaped like a saddle rack. The gooseneck area is also where we store all the stuff you need when you take horses some place. Electric brakes and tandem axles. Single door on the back (where the horses are loaded) and a door (human size) on each side in the front. Plenty of tie off hooks inside and out. See attached picture. We had it built to our specifications, e.g., pointed gooseneck (truck is a short bed), single door in the rear (default was two), two doors in the front (default was 1, you can never have too many ways to get out of a trailer), saddle rack water tank, interior lining (default was no lining), a swing down 'door' on the gooseneck storage area (so stuff couldn't roll out into the trailer), clear floor on the interior (wheel wells do not intrude), inside lights, etc. A Google of "Collin-Arndt" will result in hits for dealers and other pictures and specifications. Collin-Arndt also makes horse trailers from mild to wild as well as bumper pull models.
Stock trailer vs Horse trailer - The trailer is primarily used for transporting one of our three horses to my wife's riding lesson about 20 miles away. Occassionally it's used to transport two or three of our horses so my wife and 1 or 2 friends can go on a trail ride. In other words, we do not regularly take horses places where we'd have to stay overnight, i.e., no need to 'live' in the trailer. We bought a stock trailer because it was cheaper than a horse trailer and more open, i.e., the horse didn't think it was walking into a black tunnel. The trailer has a divider across the middle so we could transport a total of 4 horses, 2 in the front, 2 in the back. We've had as many as 3 horses in the trailer.
The middle divider can be secured both open and closed so it's possible to haul 16' items, which I have done. I've also kept hay in there on a temporary basis.
Any questions, just holler. BTW, look for a manufacturer whose a member of NATM.