Bent Door Hinge & Partially Broken Frame

Resetting the hinges so the door sits a little further from the stop will work. Sometimes it is easier to remove the door stop and reset it away from the the door a bit, especially if you are refinishing the door jamb anyway

First you should be aware that "tempered" covers a range of glass properties. Some tempered glass is toughened where the pane is significantly stronger than ordinary annealed glass. However, when it breaks it does so into large, irregular pieces. Other glass is tempered in such a way that if the pane is broken, internal stresses reduce it to tiny pieces. This type is known as safety glass.

Panes that have their edges protected within a frame are typically toughened glass, and the stuff is surprisingly tough. Panes designed with exposed edges (such as shower doors) are typically safety glass. This is because edges are the weak point and a pane with exposed edges is much more likely to break. I recommend you talk with a glass company before proceeding. Broken panes can, and do take lives (usually from people running into closed sliding glass doors and bleeding out).

Concerning option #4 there is a product called Veranda. Home Depot sells it. It is solid vinyl dimension "lumber" that is worked like ordinary lumber. It is durable, easy to clean and 100% waterproof. It will make a good looking frame, and will bond to the glass using silicone caulk. You can securely fasten stainless steel hinges to this frame. The finished look will depend on your caulking skills. Sounds like a fun project. Good luck.