I wanted to satisfy my mechanic soul within today and I decided to renew the small parking light bulb of the right headlamp as it became dead. While I was trying to mount the lamp unit to its place, the screw and the metallic flat bearing beneath it, just dropped into the engine bay and got lost. Should I remove the plastic cover under the sump? Would it cause any damage to the engine if just ignored?

there is a good chance it will be on the sump tray like you say, i very much doubt it will cause any harm if you leave it - it will probably fall out eventually. but i would try find it if you can

I did this with screws and bolts, and they just fell straight out after a few miles, it was fine...