Trying to understand the causes and implications of kinks in fatty acid chains
No, trans fats rarely occur in nature, and are generally thought to be very unhealthy.
No, like all neutral atoms, hydrogen atoms attract each other, albeit weakly (by London Dispersion Forces). That's why neatly stacked saturated fats are solid. The kink is to do with the structure of a hydrocarbon chain; their structural formula is usually drawn as a straight line, but it's actually more of a zigzag, each carbon at an angle to the last. Loosely speaking, the double bond interrupts that, so you get one more zig than zag, resulting in a kink.
See 2.
They don't need to straighten out to become solid; they just need to attract each other enough to overcome the tendency of warm molecules to keep moving around.