Can helicopters fly upside down???

Yes, But mostly in theory. A conventional helicopter will be too much unstable (Think of an inverted pendulum) and will require special mechanics (like capable rotor-blades and swash-plate). Most Combat helicopters can fly inverted for a short period of time with their engine power, design, and capable pilots.

Not, the helicopter is possibly the most unnatural flying machine, actually is hanging of the rotor, and what is hanged, is under the knot, not over it.

We can find several videos (youtube) of copters turning upside down for a breve time, but you must notice that in the moment the rotor is almost under the helicopter, it start to fall, it is not flying at that point, is free falling over its weight and have to complete the turn to avoid the crash.

Again, rotors are trowing the helicopter that hangs under them, if the rotors pull from down the copter is most like in accelerated free falling than flying.

Some helicopters can do it for a brief moment. I've seen it at an airshow.

THEORETICALLY, yes, for UNLIMITED time. however, for practical reasons, this option is limited to 3D aerobatic RC controlled MODELS because this feature implies negative pitch for the rotor, which is NOT used on big helicopters that are SUPPOSED to have their bellies hugging the terrain rather than perform fancy aerobatics in the SAM kill zone.

plenty of helicopters with fully articulated rotor hub can do a barrel roll.
helicopters with rigid rotor hub can do zero maneuvers and the likes.

Apache, Tiger and Lynx are a few attack helicopters that can do a loop.
however, helicopters as big as CH-53 demonstrated a barrel roll, understandably with a loss of altitude

No, but some kinds of rigid rotor helicopter can roll inverted for a brief period

Some combat helicopters can go completely upside down for brief periods of time, but they can't stay that way.

Basically the rotors would be pushing the helicopter down, so if the helicopter starts high enough it can do a full roll but it will lose tons of altitude and will crash if it doesn't get upright very quickly.

No because a helicopter rotors push air down so it can lift. So flying upside down would actuly make it rush twords the ground

I don't know if they can but if it's possible it probably won't be able to stay upside down for a long time without losing control.

Germans have models that can do that - it's one of those weird things only Germans can make come true : like a 5 cylinder engine !
The rotation of a crankshaft is 360 degrees - so, how can you divide 360 by 5 !!?

Some helicopters can fly upside down and other helicopters cannot fly upside down.