What foods to cook in a new cast iron skillet that will help improve the seasoning

So, just answering the new part of the question:

You can simply season "over the top" of the existing seasoning. To do this, you'd go through the normal seasoning process, wiping the pan with oil, heating in a 500F oven, wiping again, heating again. This will fill in the "holes" in your seasoning and you'll be able to use the pan again.

However, you will also end up with somewhat rough, uneven seasoning on the bottom. This will make things more likely to stick than if you completely reseasoned the pan. You can improve this by scrubbing the bottom of the pan with a mixture of course salt and oil. This will "sand down" the seasoning without completely removing it.

You should also ask yourself if maybe the original seasoning of the pan wasn't faulty, given that some of it already scrubbed off.

All my attempts to "clean and repair" a sticky-seasoning layer have ended badly as well. I know who already snickers in the background, but my opinion is: start clean. Give the pan a lye bath and a good scrub, then reseason with a better seasoning.

"The new hotness" of flaxseed oil goes back to the fact that it doesn't go rancid easily, without heat it just dries out making a film which can protect things like wood. I bought into it, and seasoned a pan with it, and wondered why all my seasoning tries were unsucessful (I stripped and reseasoned 3 times, I think). I came to blame it on the smooth forged iron.

Then I got Cookwise and read the chapter on fats. And surprise, what does it say? That PUFA stick to everything around. (Makes sense, if you stop to think about it - those unsaturated bonds are unstable, they itch to break apart). So, when I got my next iron pan, I seasoned it with three (very thin) layers of flaxseed oil (for a solid base which will stick to the pan and the finishing layer) and then two layers of lard. After the oven, I heated it from brown to almost-black on the stove (empty).

This seasoning turned out perfect. I made crepes on it, re-oiling once every 7-8 crepes only (no fat in the batter). The pan released them like a charm. The old flaxseed-seasoned pan still sits around with some carbonized matter burnt onto the seasoning, and waits for a lye-flaxseed-lard session.

Of course, I would still take some caution with new seasoning and oil well the first few applications (I only tried the crepes after I noticed that less problematic items work great). And if you tend to often fry with the fashinably-healthy nonsaturated vegetable oils, don't heat them too much, else they could bake in a sticky-seasoning layer onto the pan. In the worst case, if you do get sticky-oil (but not carbon) buildup, try adding a new lard layer before you strip-and-reseason.