Saturated va unsaturated fat is basically about whether that particular fat is a solid or a liquid at room temp. There are multiple varieties with saturated generally being less healthy because it raises the "bad" cholesterol. They don't change from one to the other with temperature, though some can change with chemical alteration.

Trans fats in cooking MOSTLY come from plant based fats that have been altered to be solid at room temp, like saturated fats. They do occur naturally in small quantities in some meats. They are cheap, shelf stable and for awhile it was thought the plant based fats might be healthier than animal based frying fats all of which made them appeal for fast food frying and such. Turns out they hit that "bad" cholesterol (LDL) worse than even natural saturated fats.

Cooking oils and other things to fry in are basically just pure fat. That makes them VERY calorific and if your fried food picks up much of any of it it will have more calories in turn. Carbs also crisp nicely when fried so you tend to see a lot of added breading in fried foods or frying of inherently starchy things like breads and potatoes which are already pretty calorific to start with.