Remove the shutters.

Consider having a roof fabricated from copper to coordinate with your bay windows. I am assuming that your leaks are a consequence of the entrance having a flat roof? Sorry to hear that....

A flat roof needs enough slope to avoid water issues. I would have the overhang rebuilt to mimic the design you have over your lower windows. Use two simple posts instead of the columns, and incorporate an overhead light if possible.

Yes a flat roof does not promote drainage. What would it look like to coordinate with bay windows? Would a railing above not match the style?

There is no point to having a railing above when you can't walk out on it, or have long windows that look like they open up. I'd find someone that can do the curved top just like your windows above the door.