I would maintain the deck that is directly under your existing eave (outside the door) at the current height as it could lead to a whole other problem with the brick siding.

And than have a wrap around step that goes down the the main entertainment space: increasing the size of the deck to encompass the empty space between your existing deck and the house (shown in the last photo).

With a single step down from that to ground level.

Currently it is hard concrete and brick directly under the eve at the door, barely large enough for a chair, it then flows seamlessly to a same-level deck. (You can make out some of this on the 2nd photo).

If we stepped down from the eve level at the door to a concrete slab, what, if anything, should we do about a roof or coverage?

Pergola for the roof, with or without glass top or any wisteria or other vines growing atop it. Preferably wrap it around the corner of your house to form an "L".

http://www.patiocoversolution.com/wp-content/uploads/2013/05/Pergola2058.jpg

http://www.westernredcedarpergolas.com/images/682_pergola_002.jpg

https://d10k7k7mywg42z.cloudfront.net/assets/547380f2c0d6715e420135d9/AUS_010_square.jpg

Adding a glass top to part of it would enable you to add ceiling fans, should you choose to do so, and enjoy a light summer rain from the deck, too. One corner of the "L" could be a place to park your covered bbq grill, too -- handy but not where you sit most.

With a sturdy frame, you can add hanging baskets filled with strawberry plants or ivy or flowering plants to enjoy and even to help shade your eyes.

Very much like the birdhouse. More posts in the yard with hanging brackets on all four sides could hold baskets of growing plants, sun catchers, wind chimes and humming bird feeders, etc...

You should at least consider retractable screens. They will solve the privacy issue without obscuring your view. Depending on how you want to use that particular space, a permanent roof in conjunction with the screens turns the area into a full 3 season room protected from wind, rain, sun, bugs, and neighbors. The retractable convenience allows you to enjoy the open air anytime as well.

Here is a link to our site:

Retractable Screens in less than 1 minute!

Ask yourself how important this area is to you and how do you intend on using it.

Other options would be a pergola with the screen(s), or a retractable awning with a drop shade/valance (it's the same screen material that extends vertically from the front bar of the awning upon full extension. The drop shade is typically used for privacy, early morning and late day sun).

Good Luck,

The Shady Folks at Garden State Shade!