My first thought was to sandwich the mesh between a couple of boards
There's no need to invent something new here. All the screen doors I've ever seen have the screen wrapped around a frame that's then inserted into the door.
There are plenty of articles and plans about screen doors freely available on the net, so search around for a plan that you like. Here's one from This Old House, for example. Once you fine one, you can adjust it to suite your situation, such as adjusting the height, width, and thickness of the door.
is it alright to replace a 1 3/8" door with, say a 1" or 1 1/4" door?
It's your house and your door -- your opinion (and perhaps your spouses and/or landlord's) is all that matters. Most likely, if you paint the door to match the trim, nobody is going to notice if the door isn't quite as deep as the jamb. Maybe the next inhabitants will look at it and think that it's not quite right.
You'll want one face of the door even with what I would call the face of the door jamb (the frame that encloses the door), and you'll need some narrower hinges. You'll probably also need to add a 3/8"-thick filler strip around the inside of the jamb so that the door has something solid to close against. But as long as you do that, the thinner door should work fine.
I have access to a drill, jig, and router + anything that Home Depot may rent to me.
Making good, clean cuts that are perfectly square could be a bit of a challenge with those tools. You could make a rough cut with a hand saw or jig saw and then clean it up with a router, or you could rent a job-site table saw. Being able to mount the router in a table would be helpful, but it doesn't have to be anything fancy.
Screen doors are typically made using cope and stick joinery. You don't have to do it that way -- you could use mortise and tenons, or dowels, or some other method -- but a cope and stick joint will be easier to form than a mortise and tenon, given the tools you have, and much stronger than a butt joint. Special router bits like this one are available. They're not cheap, but it's the right tool for your job. Cope and stick joints are stronger because they have some mechanical interconnection, and because they provide more surface area to be glued. And the best thing about this style of joint is that it makes it easy to create a very polished look.