Im not sure a 4X4 will hold a swing door 5' wide. Seems you may need something more substantial, even if your putting this in concrete. It is alot of stress.

I plan to attach/brace one 4x4 to another one just next to in on my neighbors fence and the other will have a 36" gate for my sidewalk. I was hoping this my help balance the two.

If your ground isn't too rocky, dig as deep a hole for your 4x4's as you can manage with your post hole digger. A good gravel base at the bottom is always a good idea. If you use concrete, screw some good size screws in the submerged part of your post on all sides and let them project enough to give the concrete something to lock onto (and still fit your post into your hole). You don't have to use concrete. Gradual misalignment can be corrected. Your concrete laden post might be less likely to shift, but it does get misaligned, it will be a lot harder to re-position.

I'm just an amateur but I've done a few of these. One thing I would strongly advise is to build your gate in the opening you are going to use it in. The first time I did this I took measurements and made the frame somewhere else, when I tried it I found it to be a poor fit. Instead get a helper and make a level platform at the height above ground you want your gate to be. Cut the parts elsewhere but assemble your frame right in the opening, it worked a lot better for us. A diagonal brace for the door would be much recommended I think.

"My tendancy is to want to lay the 2x4's flat against the pickets but this seems less common. " Vertical 2X4s give more support against sagging, less against warping horizontally. For fences, some use the top 2X4 horizontally and the lower one vertically, but that may be less desireable on a gate.

As with putting up a building, the foundation is quite possibly the most important. I helped relace 2 gates on the grandparent's farm around 1983. Both of the old ones were built in the early 30's. We followed the same basic layout. For the posts that anchor the gate(s) itself, we dug a hole about 3 foot deep & used a telephone pole that was sawn to fit. The pole was around 12 inches wide and we dug the hole a bit over twice the diameter. We did not use concrete but rather shveled in some dirt, compacted it with a twobuhfour, added more dirt, more compaction, more dirt, well you get the idea. Now this was a single gate that led to the pasture and was used every day. The gate itself was made of galvanized steel and holds up to this day.

"My tendancy is to want to lay the 2x4's flat against the pickets but this seems less common."

That is exactly how I made mine. The only exception is I made the box with a simple lap joint at the corners to provide a stiff joint with a lot of gluing surface. I also braced the box with a 2x4 inside the box from the latch to the lowest hinge. The frame for the gate outlasted the pickets. I could have simply replaced a couple pickets and it would have been good as new, but after 10 years I was sick of looking at it. So I replaced it with a welded steel and wood gate. It allowed to expand my skills and add welding.