I see your dilemma! I suggest that you consult with a trusted landscape architect who can help you with the hardscape and with your drain issue. They will charge a fee, of course, but doing this wrong could be very expensive.

Agree get some expert help. Our mason who laid our brick walk ran the drain pipe under the walk to a stone well area (not sure that is the correct term) out in the lawn. It works great. If you widen the stairs then widen the walk the door will be more visible

As for the gap to the right of the steps, I see two possible solutions. One is to make a narrow brick raised bed in which you could plant something. The second is to put a trellis on the wall and grow something skinny, such as one of the clematis cultivars that will grow in shade. This would involve some research, but 'Betty Corning' might be one of them.

Now, as for the downspout, I see the solution as going underground under your sidewalk and out to the lawn. I have two of these, but mine don't go under any concrete. It won't require anything underground in the way of a drywell or something, it just needs to have a pipe that goes up to the lawn with a little drain cover that you can take off to remove any debris like seeds or leaves that get in there. If you go this route, you will have to periodically inspect your gutter to make sure that nothing is accumulating up there that might clog your underground drain. As long as you do that, the system will work quite well.

I agree with the other posters that you need some professional help with this.