you are on track with your inspo.

use the same door style on your closet doors that you choose for the front door.

If you raise the entry to the upper level, you won't have any steps inside the house and you'll gain all foyer space back.

i like that Idea of raising the entry, but I fear it is beyond my budget.

You are correct that the current rail, closet door, front door and light fixture don't suit the style of your home and appear to be cheap builder-grade renovations made by a previous owner for sale. Your choice of a horizontal simple black rail is perfect. Your choice for a light fixture is a great improvement for such a high-ceiling area. You might want a black downrod and an integrated LED if you don't want to change a bulb.
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Regarding the doors, since you really want a storm door, I'd start by selecting a contemporary storm door. Look at the options out there for contemporary storm doors: https://www.homedepot.com/p/Grisham-Nuevo-36-in-x-80-in-Universal-Reversible-Hinging-Steel-Black-Security-Storm-Door-37221/330414100
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Long term, your home would look so much more sophisticated with black window frames (no shutters) in a charcoal bodied house. The white replacement windows make me so sad, because windows are expensive. Keep your slate-look patio and take care of it.
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So, with that in mind, paint the door whatever color you like or try the charcoal grey now that you would choose for the siding later, like SW Peppercorn. Paint trim SW Tricorn black, not the current white.
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Once you choose the storm door, you can choose a front door that won't clash. You have a sidelight, so a simple slab door (no raised surfaces or window) would be the cleanest option. And once you choose the slab entry door you can choose a similar plain door for the closet. You seem to prefer sliders. You could consider frameless mirrored sliders if you want to bounce more light into the entry. It would give you a place to "check your look" on the way out the door.

Entrance: Sounds like you want a storm door to allow for more light, but if you choose a front door with multiple windows, you shouldn’t need it. It would also be a shame to cover and with the money saved, you could upgrade the door/add nice trim. Suggest removing the shutters and if you have a garage, be sure its color works well.

Closet: instead of sliders that can only open one side at a time, consider this style and black would be a great choice to complement the staircase.

Staircase: lots of options with the horizontal railings and if you can’t decide perhaps the cost variations will help you decide.

Accessorizing: Regarding the interior and exterior lights (and address) you’ll have a much better sense when the above items are all completed.