I think trying to remove the rust either physically or chemically risks removing the letters as well. Also, sealing the rust will preserve the patina of the piece. But start with light coats of sealer after testing in a less visible area, A thick coat can darken the color quite a bit. This might enhance the lettering or obscure it so better to proceed with caution. Also, you might consider adding a sheet of thick glass to make a top if the sealer doesn't do the trick. It would protect both the luggage and the lettering from each other.

Hey - did you manage to fit it all in?

How do I see a very small bathroom without toilet?

You say the kids come in the small door and then bring their stuff to the large door to drop it? I think the "mudroom" needs to be at the small door. Anyone who brings their stuff out of that area has to clean the glass doors next time they need it. As for the guest entry, you need to get something bigger than that desk, and your picture needs to be lowered so the center point of it is 5' above the floor (you'll have to calculate the centimeters.) Consider making the investment to reverse the swing of the door so you can make the best of the larger wall to greet your guests with a console or credenza and some decor.