New paint cracking on caulk bead along exterior trim and j-channel flashing

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.

If you're going for maximum lifetime, you should make sure to prime all cuts with an oil based primer. If the lumber didn't come pre-primed, then prime all surfaces with an oil based primer. If you use a fast drying primer, you can get a coat of paint over the cut before you install it. I wouldn't worry about using a "sealant" between the boards. I'm guessing you mean "caulking" when you say sealant.

You should install the trim, and then caulk the seam between the two pieces of lumber after the trim is on the wall. Let it dry over night, and then apply two coats of paint over everything, including any caulking that you've applied. One mistake I see fairly often is that people will caulk something to try and keep water from getting in, but then they don't paint the caulking. The sun dries it up, it shrinks just enough for water to get behind it, and then the caulking has the opposite effect. It actually traps moisture behind itself, slowly rotting the joint that it's meant to protect!

The best thing you can do for yourself in all of this has nothing to do with paint. Make sure you buy solid lumber. Do not ever use finger-jointed pine in an exterior application. I have seen so much rot over the years, because people use finger-jointed lumber in high moisture environments, and it just falls apart in a decade or less, even when it's been painted. That stuff is for interior use only!