I disassemble the car engine. This would be most natural word order to me. No need for a preposition. Usually when using 'disassemble' it is the whole item you take apart. I remove the engine from the car. For this verb, a preposition is need.

"I removed the engine from the car." would work better. You disassemble or dismantle something, no need for a preposition. I will disassemble/dismantle that Ikea table tomorrow.

I removed the water pump from the car and disassembled it. I disassembled the control panel from the copier. <-- sort of works I disasembled the control panel while still attached to the copier. The disassembly of the car was really only removing the head. Can't see any grammatic problem with the likes of "I disassembled the hands from the clock." but removed sounds better.