I think you've made a wise decision to use trim. Wood expands and contracts mainly across the grain, and almos nothing with the grain, so expansion and contraction of the trim isn't an issue. If you go with the 6mm ply panel, you won't have any expansion to speak of, however, you can make certain by gluing the panel in place only in the top and bottom center of the panel. It is being held in by the groove anyhow. Whatever finish you decide to use, finish the panels before you assemble them. Very hard to apply finish thoroughly after assembly.
Once you glue on the trim, with modern glues, it will NEVER come off. Get the slowest setting glue you can find so you won't feel rushed. In a warm room, the most common glue sets up in 2 to 4 minutes so you don't have time to fiddle with it. The slowest setting glues stay "open" for 15-20 minutes. Use a small brush or a small stick to apply the glue to the door. Don't use very much glue because when you place the trim, it will squeeze out. You can't apply finish over glue or you'll have a white spot. If glue does squeeze out, you let it sit for a few moments and as it stiffens up, scrape it off. A little sanding after the glue is dry will remove any residue.
I suggest you place the roundove trim starting on one edge and work your way around the door, fitting one edge at a time. It is easier to make if fit and look good that way. You do this fitting and cutting before gluing, BTW
To hold the trim in place, I suggest you consider borrowing or renting an air compressor and buy what's called a pin nailer. This tool is very inexpensive and shoots in very fine pins with no heads. They make a tiny hole that is almost impossible to see (remember you can fill cracks and pinholes by rubbing a bit of Timber Mate filler across it). Buy the Timber Mate from Amazon, and order the color that matches the wood you're working with on the trim (probably a good, kiln dried pine).
For a smooth surface, I start by sanding with a 150 grit paper, then up to 220. Any finer grit makes particles that stick on the wood grain. Before you finish a sanded surface, buy a package of waxed "Tack Cloth" to rub on the surface and remove all sawdust. Almost any paint and finish materials store will have this.
If you want it to look best, apply a sanding sealer to the wood before sanding. This raises the grain slightly before you sand and renders a smoother surface. Apply and sand within two hours, but let it dry a bit before you sand.
Most of the time, I finish using a simple combination of water-based stain in whatever color I like best. On the clean, clean, dust free project, I apply some pre-stain to the surface, which smooths out the stain's coverage. Within 2 hours, I then use a cloth to rub the stain lightly onto the wood and wipe off any excess. Stir, not shake, the stain gently but thoroughly to make sure the particles in the stain are mixed thoroughly. Apply and repeat until you get the color you want. Try this all out on a piece of scrap first and take notes.
You can get water based or oil based stain. Oil based stain must be applied, the excess wiped off (reapplied for darker finish, wiped again) and allowed to dry at least 24 hours before applying the final coats of finish as described below. I really prefer the water based stain. If I forget to use the sanding sealer, after the first application of stain I must then sand the raised fibers off with a few light passes with 220 grit sandpaper, then use the tack cloth. This happens on occasion because I'm human.
Finally, I most often use something called "wipe on poly" in either gloss or semi gloss. This gives it a shiny appearance and seals and protects the finish. I apply this with a disposable brush or just as good, with a paper towel folded over several time to form something like a brush. You can thin down the wipe on poly to give it a longer drying time and make for an even smoother finish. Instructions for thinning is on the can or bottle of poly. I usually apply 2 to 4 layers, until it looks perfect.
The finish really makes the project look either great, or anateurish. Worth taking the time to do it right.
OR, you can sand as described, tack cloth it, and apply paint. Simpler and paint works unless you want the look of finished wood. I made a 10 foot long set of cabinets topped with two book cases and a large space in the middle for the flat screen TV. It is painted because I always like the look of painted book cases, and the plywood I used didn't look all that good to me.
Hope you didn't get bored with my short novel of a post, but finishing is really important, and most of finishing is about the preparation of the project. My method is one of MANY ways to finish a project, but it has been easy and reliable for me and doesn't cost much.
BTW, I use small, shallow plastic food cups to hold the stain and poly. Don't dip your brush into the can! You will likely add some particles to the stain or poly that will ruin the whole batch. I use disposable plastic spoons to ladle out enough of either to do the job.
The more I do, the less I accomplish.