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.

Let me know how that turns out! Sounds interesting.

Well, I always use homemade pie crust, which I roll out then cut into a 7″ circle (I use the lid of a plastic pitcher – it’s easy to hold and sharp enough). I fill mine with homemade pie filling and then fry it in peanut oil. If I were to bake them, I’d likely do the same thing. So let’s see. If I were using store-bought pie shells, I’d still have to roll them out and cut them smaller, so I’d probably use the same top of a pitcher or something similar. Then you could either use an empanada press, like I do (here’s one at Amazon: ) and put the filling in the center then bake it or use tart pans. Honestly, it’s easier to use the press. Put a little egg wash around the edges before you crimp it in order to seal it.
 
I wanted to work on my recipes (particularly my dough recipe) so when I first started out I was a lunch wagon at a local plant and I put a bunch of things in the pies – bbq beef, chicken-bacon-ranch, meatballs in marinara, taco meat and cheese and, in the mornings, scrambled eggs and cheese. They went over very well and I think they’d be good for you too. I now only use fruit filling because that’s what sold best. You could keep them warm in a small oven.
 
To be honest, the hardest part for you would likely be the dough. After all, you’d be buying something that’s already made into a 9″ circle and you’re having to re-roll it and re-cut it into something smaller. My dough is special because I fry them and you can’t use just pre-made dough or it falls apart in the oil. You have to have some kind of protein (like evap. milk or buttermilk or egg-and-vinegar) to bind with the flour. But if you’re baking them you should be able to just use regular pie crust.

It’s the narrow “mouth” that I was thinking would present difficulties, but mostly it was the fact that it wasn’t a sealed unit but an open design that gave me pause.