Can we say "Punându-mi mâinile pe cap, eu am început să plâng." instead of "Punându-mi mâinile în cap, eu am început să plâng."? The în sounds really strange to me.

Actually both are correct in Romanian. PE CAP would be taken rather literally as putting your hands on the head and IN CAP should be treated more like a sign of desperation.

So the police will ask you to put your hands PE CAP, while your mother will put her hands IN CAP seeing the mess in your room :)

I think both are correct. In usual conversation in is used more often than on

It sounded very odd to me too, but if "în cap" is a colloquialism, then you can't really change it to "pe cap" just because it sounds better.