Follow up from OP...
First thanks for all of the great input and suggestions... especially those relevant to my questions.
A few notes...
- I chose the 120V lights because I got a really good deal buying stringer and bulbs seperately and I can replace them individually one day when they eventually die. My understanding was that there were 120v LED bulbs and 12V LED bulbs but maybe that was just marketing. I do believe the stringers themselves are wired differently. I could not find the 12V bulbs in bulk, just individually for about $2.50 each. It probably would have been a better technical choice to go with the 12V strings. I'll check out the link provided above if I decide to change anything.
- My battery is almost brand new. And I determined that was NOT the problem. The problem turned out to be the one unpredictable variable (yeah me). When wiring everything together I used several crimp-on quick disconnect plugs. And though I was smart enough to mark the positive and negative ends, I wasn't smart enough to pay close attention the last time I put it all back together with the charge controller. Fortunately the controller was smarter than me and noticed I had the wires crossed... and every time I connected ANY load it immediately killed it. So now I'm back in business.
- I did find an inverter without a fan... so as not to put any unnecessary drain on the battery. I'm also going to replace the fuse with one rated for 10Amps. That should be plenty for my needs and might also save the controller one day.
I don't think I mentioned this but the plan was never to build a major power center. I don't need to live there or operate power tools (though that day may come). I just need to see inside my shed at night occasionally for a few minutes to grab or store something, and it's usually awkward to juggle a flashlight. The LED bulbs seem to do a great job, I have the spotlight if I really need to get into the deep corners, and the 7 Amp Hour battery holds up quite a while longer than I ever need it to.
I think that's it!
Thanks again all.