Good on ya for taking the plunge on DIY -- it will soon occupy much of your time, effort, and at times money
I can't really offer definitive advice on your specific project, but will try to give you some feedback from my experiences rebuilding our baby. Many projects were done while completing the Loop, so I can fully appreciate using the cockpit as a workshop.
I'm having a hard time visualizing your design, so no thoughts there.
"Now the hard parts... what do I need to know about working with teak?"
Nothing mystical about teak. It's a harder wood than pine, but it's still wood ('tho a lot more expensive.) Something to keep in mind is that teak's an oily wood, so always give it a good rub with acetone before you apply any finish -- if you don't, the finish won't get a good bite on the wood and it will fail prematurely.
Small hand tools work fine. I can't think of anything I'd want to use the Roto-Zip for, though. Likewise, the circular saw's likely overkill. If you don't have one, a sabre/jig saw is worth the modest investment. The Dremel's a pretty verastile tool, and with the right bits/attachments can produce decent results. I took the plunge and got a Fein MultiMaster -- still amazed by how much time/effort/aggravation that baby saves me.
"The adhesives I have on hand are standard wood glue and 3M 5200 -- which would be better, or do I need something else?"
For interior work, regular wood glue will work but is not a great choice. I keep a small bottle of Gorilla glue and a bottle of 3M Titebond III aboard. Both are waterrpoof glues that are easy to work with.
BTW -- 5200 is a poor choice for any wood project. It's also a very questionable choice for ANYTHING ELSE aboard... I used it once (and only once) to glue some wood blocks onto a piece of scrap plexi to make a cheap long board when fairing the bottom of my hull. Worked great, but I knew I was going to toss it away soon.
"Should I get some small stainless steel tacks/finishing nails to hold my strips down, or would glue be sufficient? My "clamp" will probably be the latest box of stuff from Defender set down on top of the glued parts "
Invest in some cheap clamps -- C-clamps at a couple of bucks a pop are a great deal. They come in handy, and don't take up much room.
"How should I attach it to the bulkhead?"
Unless there's no other alternative, use fasteners instead of glue. It will be much easier to refinish it if you can take it down to sand/clean/finish. Nothing's ever permanent on a boat, so the easier it is to remove the more ahead of the game you'll be.
"Also, if I want to varnish my new plate rack, how would you recommend I match the bulkhead color? Is varnishing a good idea?"
If your bulkhead's teak, your new piece should match it close enough without any additional work on your part. You can check it by wiping the unfinished wood with mineral spirits to see what the finished color will be. If it's too dark for your liking, bleach it with a two-part bleach or oxalic acid.
For an interior finish, polyurethanes are more than sufficient. I usually use either Minwax Poly or Minwax Helmsman Spar Poly. Regardless of your preferred finish (gloss, semigloss, or satin), always use gloss for your base coats.