What is this plastic cord hanging from underneath the front of my car?

SOURCE: 2004 VW Passat engine cover under car hanging

dead right it is ! the cover has not been resecured properly after changing oil +filter if garage says it waslike this on service why not tell you ?? either way take it back and complain this cover is secured by screws and no excuse for not fitting securely

SOURCE: The plastic covering on the under carriage is

Should leave on and get fasteners from dealer or fastener shop. Auto body shops and suppliers will have them too.

SOURCE: Piece of plastic is hanging down under front end

It is most likely the front lower valance.It may stay in place or it may fall off depending on how many fasteners are still in place to hold it in position.

SOURCE: BIG PLASTIC PART FELL FROM THE BOTTOM OF MY CAR

Probably the engine undertray, not a vital part, but as you clearly don't know what it is then you also aren't qualified to investigate or remedy this any further. It could also be a wheel arch liner.

If you've had the car recently serviced then take the car back to complain, as the undertray wasn't correctly refitted. It does serve a protective purpose and improve aerodynamics, so it's a good idea to get a garage to replace it. if damaged they can either order a new one or get a cheaper used one from the wrecking yard.

SOURCE: the undercarriage plastic covering is dragging on

Replace the mounting bolts. You should be able to find them in any pep boys, napa, etc.