Trim separating–caulk or wood putty

Sure. Why not? If you have a brad gun with headless pins, they leave a hole that would be almost undetectable from a few feet away anyway. With a little care about where you place the pins, you might even skip the step of filling the holes, certainly so those in baseboard and possibly in ceiling trim.

For trim at eye level where it may be more critical to hide the holes though, the right color wax crayon used for filling nail heads will do wonders. You can buy them in wood colors for this purpose. Rub a little in, then clean the surface with a cloth or your finger.

The strips you made and installed are called extension jambs. We use them all the time to fit new windows and doors. I usually sand and prime them before I install them, but if you have already installed the jambs and old trim, here are a couple of ways to finish them off.

1. Put a straight piece of blue painters tape down the wall adjacent to the door trims.
2. Sand the edges carefully so the jambs and trim are flush.
3. Clean off the dust and apply a bead of latex/acrylic painter's caulk.
4. Using a 1" putty knife, smooth the caulk to a nice flat layer against the jamb/trim.
5. Remove the tape immediately. Retouch the caulk gently if removing the tape leaves a ridge.
6. Let the caulk cure for 24 hours. Minor imperfections can be smoothed out with a sharp utility knife and a damp cloth. If there is some shrinkage in the larger gaps, add a little more caulk, smooth off with your putty knife, let cure.
7. Paint.

There are painter's caulk that will "fast dry" in an hour, that you may consider. If you are unhappy with the final leveling of the caulk, you can also use a super thin layer of lightweight spackle (not standard vinyl spackle) applied with a putty knife over the caulk. This product dries in about 20 mins and can be lightly sanded or shaped with a damp cloth.