How do you fix loose brass inlay?

Re-bedding brass stringing requires animal hide glue. Modern adhesives do not stick well to oily rosewood or smooth metal over time. Clean the channel carefully. Apply the hot glue, press the brass strip down, and clamp it with a flat block until dry.

I have a writing box that is locked shut and the original key is missing so what is the best way to get it open without breaking the wood?

Take it to a specialist locksmith who handles older furniture. They carry a ring of blank keys and can usually pick the simple lever locks used on these boxes within a few minutes. Forcing the lid with a screwdriver will crack the veneer and damage the lip. Sometimes you can gently tap the side of the box while lifting the lid if the catch is just stuck. Picking the lock is the safest route.

What kind of oil cleans rosewood?

Avoid oil entirely. Oil darkens the grain and attracts dust. Stick to a good microcrystalline wax applied sparingly with a soft cloth.

My box has a piece of leather inside that looks really dry and is peeling away at the fold, should I try to glue it back or just leave it alone?

Leaving it alone prevents accidental damage. Dry rot in leather hinges makes the material brittle and powdery. If the leather acts as a functional hinge for a writing surface, you might need it replaced by a desk restorer. Tucking loose edges down with a tiny amount of reversible paste is fine. Do not flood the area with modern glue.