You're right - it's not rocket science, which is making me feel like a complete idiot.
I figured out the first problem I was having - I wasn't using enough batter. But I don't see any websites addressing the issue of refrigerate the batter vs. leave at room temperature, or what's causing all the little brown bits.
I have a propane flame under the cast-iron, flat-bottomed large oval dutch oven in which I have about 6" of peanut oil. It's able to keep up the heat. Perhaps you're correct, Cheeseburger, that the batter's too runny. I was using the Gold Medal Funnel Cake mix but I'd like to make my own from now on.
The best results I've had while pouring is to put it into an actual funnel with one finger plugging the hole (and thanks to Sirloin and Double Chili Cheeseburgers for the videos -they helped). Filet Mignon - I think you may be right about me being too far away from the oil but I was basing it on the videos and even when I was pouring right next to the surface of the oil it still gave me a ton of little loose strings of batter that floated off.
Sirloin, you may be right that I'm ignorant about this stuff. I watched the videos and honestly didn't see anyone skimming, at least to the point that I'm having to skim. Seriously, I'm having to spend at least five minutes trying to get the crud out of the oil with a very good, very fine-meshed skimmer.
And Filet Mignon - if you're ever in the Ozarks please stop by and I'd be happy to fill you up with some fabulous fried pies. I make 'em better than anyone else (according to my customers), which is why I'm feeling frustrated that something as relatively simple as a funnel cake is kicking my behind!
So I'll try thickening the batter and make sure to hold it close to the surface while pouring. If anyone hears about whether or not to refrigerate the batter or if that's what's causing my issues, please let me know. Meanwhile, I really appreciate your advice and links. You're all that's keeping me from giving up in disgust.