Jim,
I really like your project. Especially for a first time try, you did great.
As far as the narrow part breaking off, well "poo-poo occurs" for all of us. I probably would have saved the broken piece and glued it to the backer in the appropriate place.
As far as staining goes, I never stain or put any finish on until the whole part is cut out. I seldom stain the top piece anyway and the backer is OK to stain (in my book) to get the separate distinction of color but, do not put any oil or coating on until the gluing is done. Glue doesn't like to stick to oil or finish very well. You can use a Q-Tip, tooth pick, very small brush, or something to get the stain down onto the edges of the cutouts if you really need to. I don't "dip" finish at all because it usually takes longer to dry out.
I usually wipe on a coat of BLO (boiled linseed oil) to pop the grain on the wood as the last thing and maybe a coat of shellac, lacquer or polyurethane on top of that depending on the look or gloss that I want.
Scroll saw items are not used or handled much for the most part so almost any finish (even none) will last for a long time.
The best tips I can give you for focusing while cutting.....
Your eyes should be on the point where the line meets the blade.
Feed slowly and let the saw do the hard work.
Take a deep breath before starting, shrug your shoulders a bit and start cutting REMBERING TO BREATHE WHILE CUTTING! People tend to hold their breath when trying to concentrate and that is a bad thing! If you find yourself stopping a bit to take a breath, you are not breathing when the blade is working which is the most important time to focus.
Good luck with your next few projects....Oh yea, KEEP THAT FIRST ONE FOR THE REST OF YOUR LIFE!....Stay in touch and keep making saw dust. And practice, practice and practice.
Rog