Hi rocky100370,
Welcome to the world of door building. There are a LOT of ways to execute what you're looking for, but here are some ideas based on your criteria:
To hit your 1-3/8" (35mm) thickness, I normally purchase 8/4 rough-sawn stock, which gives you plenty of material to face joint and thickness to final dimension. I'd avoid trying to use 2x framing stock as it will likely have a lot of knots. If you don't have ready access to 8/4 material, you can produce scantlings by gluing up three layers of 4/4 material. Either way, I'd focus on acquiring kiln-dried material, free of knots, with reasonably straight grain, and focus on producing stock which is dead flat and straight.
For joinery, I'd suggest dowels. Of course there's always options like mortise/tenon or Domino, but both require a fair amount of machinery investment and/or set-up and skill. I recently visited several window and door shops in Germany, and all of them were using dowel joinery for their 68mm doors and windows, so you're certainly not giving anything up with dowels. Rangate sells 10mm x 100, 12mm x 100mm, and 16mm x 150mm hardwood rot-resistant dowel pins intended specifically for door/window production:
https://rangate.com/products/rot-resist-dowels
Assuming you're planning on standard leaf hinges, Templaco produces affordable hinge mortising templates:
https://www.templaco.com
If you're trying to keep it simple, you can go for a shaker-style door using plywood panels.
If you're interested in a read, David Sochar wrote a nice book titled "Small Shop Production of Custom Wood Doors"