Bread – Stretchy Lebanese flatbread

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.

Yes, they are safe.

The reason why warming up is recommended is simply taste and texture:
When cool, they are more rigid, dry and crumbly, warmed up they become soft and more pleasant to eat. So if you are just hungry, start nibbling.

Regarding the food safety aspects:
Warming up will in no case make unsafe food safe, see our generic post (about meat) why even boiling won't do that. Bread is not a food that is prone to super-fast spoiling and can be safely stored at room temperature. Apart from that, those pre-packaged breads contain preservatives (see your ingredient list) and are typically sealed to prolong shelf life.