Greetings,
Let's see if I cover all these.
IndustrialMech parts on a BattleMech- Yes they can. All the limitations of that part apply. If you put a CattleMaster head on a Commando, that head is unsealed, has no ejection seat and does not have an advanced fire control system, so incurs a +1 to hit modifier.
Environmental Sealing or other add on equipment- If it is located in the part being used, that equipment still works. You will have to pay tonnage for it. Two most common examples will be the Sealing and Ejection seats.
Primitive Parts- They can be used and are treated exactly like they would be on a Primitive unit, including the critical hit modifiers.
Primitive Engines- They can be used but they provide power based on their starting power, not their modified power. So a 360 rated primitive engine would provide power as a 300 rated engine. You pay the weight for the 360 engine.
OmniMechs- It becomes a non-Omni, even if it uses Omni parts. Omni's are the Swiss Watches of the universe. They only work with the parts made just for them.
Cockpits- Short answer, game balance. Longer answer- conflicting computer system cannot be reconciled to eachother. While a Command Console has both pilots in the same location, splitting the pilots causes the computers to try and justify the locations and this is bad. In the end this one is all about game balance and any in universe reason is just created to support that.
Best,
Joel BC