For the below picture, is it a 2×4 or 1×2 on top? Also how does it attach to the pickets? Is it nailed/screwed to the posts?
I can't really tell.
Wood – Which wood dimension to run on top of fence
fenceoutdoorwood
Related Solutions
Eight feet is a common distance between posts, due to it being a standard lumber length. You will have a stiffer, longer lasting fence at 6-7 ft. An integer number of pickets with your desired spacing is a good way to decide from a design standpoint.
I prefer to sink posts to at least frost level (42" in my area). The 1/3 advice is good if frost is not an issue.
I also prefer crushed gravel (21AA or paver base) to place wooden posts (versus concrete or pea gravel). 3-4" in the bottom of the hole (beneath post), continuing up the sides of the posts to just below the surface. Give the post a push after first 5" and tamp along all 4 sides as you fill. This technique will greatly extend post life and minimize rot.
I've used the gate brackets that you build into the gate which have hinges built in. They will stabilize a 42" gate (each side). Extra long screws on the top hinge are helpful. Stainless steel cable run diagonally (with adjustment turnbuckles) run from the top of the hinge to bottom of the gate (at the opposite corner) will make it sag proof.
Disclaimer 1 user of product, no financial connection: 
Disclaimer 2: Haven't used it but sounds interesting, YMMV (Near) Instant fence post backfill
You have numerous choices to deal with the gate hinge issue. For a round post you can clamp on this type of hinge and then bolt its flat plate to the gate:
Alternatively you could drill a horizontal hole through the metal post and install the bolt type hinge pin to be mated with the type of bolt on hinge strap in the second following picture:
A third option is to use the following U-bolt type hinge point mounted on the post. Then use a lag bolt hinge pin screwed into the gate framing:
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Best Answer
That's a 2x4. It would look better on both sides of the fence if it were a 2x6 or 1x6, because then it would fully cover the posts. Nail them down from the top, into the 4x4 post.