Metals are usually used because they conduct heat better. So yes, they heat up easier, this is a desired effect since heat then transfers from item being cooled (processor) to item doing the cooling.
Copper has better heat conductivity than aluminium, at same time it is also more expensive, which is why most coolers have copper "pipes" which are then attached to aluminium fins Another matter is weight, if all coolers would be pure copper, they'd be a LOT more heavy and would pretty much require separate support along with motherboard mounting.
Heatsink as such relies on surface area, bigger area, more heat dissipates, better it is. which is why they are pretty much always full of fins next to each other. the shape itself has no real meaning unless you consider outside factors like airflow, in which case having air flow between said fins cools them down better.
said airflow caused/increased by additional fan is pretty much only difference between passive and active cooling.
Passive coolers tend to be bigger than active if they are rated for same cooling capacity (often measured by watts) since airflow helps a lot in cooling.
Additionally orientation of cooling fins is not as important in active since the fan will push the air through anyway, while in passive, heated air tends to rise up.
so for passive coolers, having fins in vertical orientation is better since it doesn't hamper the warmed up air from rising upwards where it naturally wants to go.