Are fins necessary/required for cooling in a closed room

There are two power functions you need to be concerned with: instantaneous power dissipation, and long term power dissipation. A large thermal mass (finned or not) will prevent your IC overheating due to a quick spike of power. Over the longer term, the heat needs to be dissipated to the outside environment. Fins are only effective if there is some airflow to remove the heat - this could be by simple convection or by forced flow (fan). In a sealed case, convection will be reduced compared to free air and a fan will also be much less effective.

For your case, if you don't get sufficient cooling from just normal internal convection, I would have a metal block from the top of the heat dissipating commponents to the case itself. That way, you increase the thermal conductance to the case. If you still have a problem with heating, then you could put fins on the outside of the case, which is presumably in free air.

It depends on a few factors: what’s the maximum temp of the CPU die? What’s the surface area of the enclosure? What is it made from? What is the distance between the CPU and the surface you want to reach with this copper block? What is the ambient temperature? Do you expect solar heating?

Off the top of my head I’d say no to the copper block for two reasons: weight - even a modestly sized block of copper is going to be heavy, which will put strain on the heat sink mounting tabs - and what to do with the heat once you get it to the exterior. If it’s low power, then maybe the enclosure wall can dissipate enough heat from the area around the copper block, but if it’s high power, then you just bought yourself into putting a heatsink on the exterior of your enclosure. However, if you have a heatsink on the CPU and reasonable circulation on the inside of the enclosure (from, say the fan on the heatsink), then the entire area of the enclosure wall becomes your heat transfer area. It should be fairly easy to get a back of the envelope answer to how much heat will be transferred out of the enclosure if you know the ambient temp, the max internal temp, and the enclosure thickness, surface area, and material.