1. Botulism is nearly a non-issue in lacto fermentation. As the article mentions, salt and acid inhibit its growth. For an extra layer of protection, you can add a bit of vinegar at the start of the process to further encourage the growth of safe microorganisms. Botulism only starts becoming an issue when you have low-salt, low-acid preparations, or when you start the process like you would in traditional canning (boiling the fermentation mixture ahead of time -- getting rid of all competition for the botulism). If you cook the peppers you should make sure to pitch in a significant quantity of yeast and/or lactic acid producing bacteria.
1a. Botulism is a lot higher risk if you stabilize the mixture (the "simmering it in a pan to thicken" step) before storing it. That kills any safe microorganisms and again opens up the door for an infection from something dangerous (though, as the article mentions, it's less risky because of the accumulated salt and acid in the final product).
2. Mold requires a stable, aerobic environment to grow. Completely submerging the peppers suffices, as does using some form of air-lock, but you can also almost always get away with just stirring it twice each day.
3. Sugar can make the fermentation work better. The suggestion of fruit to achieve that is a great idea. As a side-note, mango is particularly delicious fermented on its own (halves or other large chunks, fermented just till there's a lot of CO2 bubbles, making them fruit chunks sparkly, no longer or it gets mushy).
> kills off the fermentation.
4. That has pros and cons. Fermented foods "keep" better while still fermenting (in terms of food safety), but the texture and flavor can degrade (it varies, look at how fantastic kimchi is as an example of longer fermentations being beneficial). Once hybrid strategy if you still want the improved texture from cooking is to reserve some of the pre-cooked sauce and re-introduce after cooking, re-innoculating it with yeast and bacteria. It'll tend to keep a lot longer if you do so. The flavor changes can be good or bad. It'll build up pressure over time though, so don't seal it air-tight.