You'll get a wide variety of answers to your basic questions. I would follow Kevin J's advice for the most part - he and I would have no trouble making doors elbow to elbow.
I have used stave cores twice in 26 years, and one had problems (more later). All our doors, several thousand, interior and exterior, most of them with a 5 year warranty, have been made of solid wood (sometimes laminated for thickness with 2 or 3 plies), with coped sticking and mortise and tenon joints in the classic style.
Until recently, we had one exterior door give us problems and two interior doors we had to replace since 1990. Why? we know our lumber, buy from a good vendor or two, and treat it right. We make sure our customers know what to do with the doors, and we like to fit and hang and hardware our doors, as well as finish them.
The recent problems I mentioned above all stem from using Titebond III for gluing panels to width. We have had to repair or replace about 5 panels that had glue joints open up due to the fact that TBIII looses strength at higher temps - like 140 to 180 degrees. Losing 50% of its strength or more. We have glued lines open up and in a couple of cases, open all the way thru. The doors all had a dark finish, low gloss, and faced due West. I could measure 190 degrees on the surface.
The one stave core that failed had a Walnut stave core ("engineered") for stiles and rails, with a 1/16" face veneer both sides. It faced due West, and was near black. The veneer was glued on with urethane and vacuumed - it was as tight as can be. But after a few months, cracks in the veneer started to show, more at the lower edge than the middle, with none at the shaded top of the exterior side. Interior was fine. No warp, bow, twist or things of that sort. We removed all the veneer and found the cores, laid up with TBIII don't you know - had let loose as the joints heated under the face and cracked the veneers. Some gaps were .085 wide, 1" deep in a 2-1/4" door. To be fair, this was not the construction style - the stave core - that caused the problems, it just happened to exacerbate the glue failure. F#%@&# glue.
Do not use TBIII. We are now using TBII, until we get ourselves back in a reasonable mood. Or epoxy. Or urethane. Or plastic resin.
As for the half-round head, we used to cope and tenon the rails into carefully laid out mortises with a generous haunch and set them at 45 degrees to the stuck stiles. Glued and clamped, then the inner radius is sawn and the sticking is run on the shaper for the curved rail, left long at top dead center. This made the sticking winnow out at the transition from curve to straight, and was never clean. So now we haunch back the stile and allow the curved rails to sit on that haunch and the sticking then butts into the stile sticking. The joint is a square edge M&T, with no copes. This cleans up the sticking problem. Then the top rails are cut to width/length on the saw, and a double spline is made with long grain bridging the joint. This is glued and clamped when the other rails go in at assembly. Most arch head doors are drawn full size so we know exactly where everything needs to go.
I would never recommend a stranded or man made core. You do not know what it is really designed for - except for the fact that it is not designed for doors. You do not know the MC of it and would have to research the performance data. It is not a proven solution. There are some European products that appear to be excellent solutions, but they are not prevalent.
I build doors on what I have observed over 45 years and from what I can learn historically. Those methods were pretty good, and we have made lots of improvements in the construction methods to help drain water, keep things glued (with a few exceptions....), and disprove all those marketing phrases coming from the plastic, metal and fake wood people about wood doors cracking, warping, rotting, bowing, warping, etc. You unconsciously repeated one of their marketing phrases when you said" How do I prevent eventual warping?" There is no reason to expect a well made wood door warping. I do expect the doors we make to outlast the owners, their children and perhaps even the building.
If this door is for you, fine. I would not recommend making a door for sale that you may have problems with down the line. You cannot make any money on the first few doors you make. You haven't asked about the frame yet. And there are many other things - hardware, finish, weatherstrip, sill types, glass, sealants, etc. All of it has to work in all kinds of weather, at extremes, and perform all the time. It also has to look good, and be secure, but easy to operate and maintain. Lots happening in a small space, more than any other wood project.
Storm door? Why? That will certainly heat up the glue. I have seen plastic plant-on door moldings on metal doors melt and sag when under a storm door. They are effectively a solar collector. You protect the door by using the proper finishes, shading it architecturally and other methods. The storm door is just another thing to get in the way.