How do I make my sourdough bread dough stretchy and not have it tear? I've been trying to make chewy sourdough bread and it's not going as planned. I read somewhere that if you mix and knead the bread a lot and avoid adding a lot of sugar and use a lot of water that should aid gluten development to make the dough chewy. Also I read that the dough should be stretchy and  if you pull two pieces apart it's supposed to be transparent. Well, I don't have anything anywhere near that.

My starter was super fluffy. I took 2 cups of starter, added about 1 cup of water with 1 teaspoon of salt and 1 tea spoon of brown sugar already dissolved. Then I kept adding flour and mixing in one direction until it didn't mix with a spoon anymore. Then I kneaded it for about 1/2 hour to 45 minutes. I tried to avoid adding too much flour to keep the water percentage up, but that's hard to do when you are kneading on a wooden board and the dough keeps sticking. I finally got it to where there was an even layer of dough stuck to the pen so the piece I was kneading didn't stick to it too much anymore, but the dough still refuses to be stretchy. I pull on a piece and it tears with a ragged edge. That's after 1/2-3/4 hour of non-stop kneading. Finally I gave up because it wasn't getting better.

I let it rise for 1 1/2 hours between a heating pad and covered with electric blanket on top. Then I kneaded it again. At first it was slightly more stretchy, but the more I kneaded it the less stretchy it got. I didn't add a lot of flour at this stage either so it kept sticking to the board. Now it's back to the original non-stretchy, tearing consistency.

Can anyone tell me what I'm doing wrong?

Edit: I should mention that I'm using King Arthur bread flour, so the protein content is most likely not a problem.
Your recipe is to blame.

Your dough is stiff and tears because it isn't wet enough.  While different types of flour do absorb different quantities of water, I think most of the issues you're experiencing are due to the ambiguousness of the recipe and instructions you're using.  Most sourdough recipes yield dough around 80% hydration (the weight of water/the weight of flour).  The  recipe you've cited relies on the starter providing all the hydration for the dough.  Doing the math, following the feeding instructions, you would have a starter that is 55% hydration. Adding all the flour called for in the dough recipe, you would end up with a dough that is about 45% hydration... stiffer than bagels (50-55%). 

While it is possible to make dough that doesn't have any added water, it is not common, especially with such a stiff starter. Also, the volume of flour would be much smaller than the volume of starter.

Try using a tried and true recipe that doesn't rely on the starter to provide all of the hydration.

Don't worry so much about gluten production and how sugar or fat will affect it.  Small amounts of these enrichments, like those in the recipe you linked to, will soften the interior of the bread and help the crust brown nicely.  Just look at brioche if you want an example of how far you can push enrichment and still get a well structured bread.