All good questions!
The hose the fireman is using, known as the "squirt hose" is connected to the water delivery line between the injector and the boiler. When the injector is on, the fireman can use this hose to adjust or "fine down" the amount of water delivered to the boiler. More water observed from the hose, more water to the boiler, less water combined with steam, less water to the boiler. The idea is to try and replace the water that is being converted to steam without adding so much that boiler pressure is reduced......it's a dance. This hose is also handy for cleaning out cinders (coal burner), sand, etc. in the cab and on the running boards.
The small door, or "peep hole" is opened and closed as for the fireman to view and adjust the fire as needed. It is also used as the entry point for "sanding the flues", by slowly pouring scoops of sand into the hole to clean soot from the flues, which will improve steaming. This is usually done when the locomotive is working hard, providing maximum draft.
Hope this helps....MDR