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Unfortunately, unless they are in container, there is not much you can do. The tree is going to produce when it is ready to do so. Here is an article that will give you more information on proper care of figs: https://www.gardeningknowhow.com/edible/fruits/figs/how-to-grow-figs.htm

This is true. But usually it is out of the soil by harvest time. There are other ways to make your own natural rooting hormone. I like to crush an aspirin tab and mix it with a little honey and just a few drops of really hot water to melt everything together. It should be a syrup-like paste at this point. You can apply to the cut stem like you would the powder.

I would apply dolomitic lime along with wettable sulfur powder. This will keep the pH balanced while the sulphur destroys what it can of the infection. Here is an article that has more information: https://www.gardeningknowhow.com/edible/fruits/figs/figs-disease-rust.htm

If you know they won't ripen, there's no reason not to remove them now, and it could make the cleanup a lot easier. Your figs most likely do not ripen because they need warmer temperatures, but also see these articles:

http://blog.pennlive.com/gardening/2011/12/figs_not_ripening.html
https://www.gardeningknowhow.com/edible/fruits/figs/figs-not-getting-ripe.htm

Generally, you will cut them back and wrap them in burlap with several layers. Mulch the root area as well.

Here is an article for more information: https://www.gardeningknowhow.com/edible/fruits/figs/fig-cold-protection.htm

The best time to prune these is in winter, when they are dormant. There is a certain method of pruning so that you keep the proper amount of fruiting wood. This information can be found in this article: https://www.gardeningknowhow.com/edible/fruits/figs/pruning-fig-trees.htm