These are aerial roots or some called Monstera air roots. They can grow from each Monstera stem or from its existing nodes to support the growth of the plant. As it grows, Monstera Decliosas are constantly reaching higher for more sunlight and that's where the air roots play its part. The aerial roots dangle in the air to capture nutrients and carbon dioxide, supplying more energy to the mother plant. The brown aerial roots can grow as up to 20 ft long in its natural habitat. Since Monstera Deliciosa live under jungle canopies, they are constantly trying to climb higher for more sunlight.

Yes you can. Cutting the aerial roots from your Monstera Deliciosa will not damage the plant and the roots will grow back in no time. You can also leave them be but it can be an eyesore to some people. These air roots can grow out of control and can look unruly cables. Just make sure to not damage the Monstera root node when trimming the air roots. But yes, keep calm and chop them off. 

If you decide not to cut the aerial roots and they are getting too long to manage, simply tuck the air roots and secure them to the stems or totem pole with ties. Or you can tuck them nicely into the pot and leave the roots to be. 

No. It will not work.

Aerial roots have no propagation function and will not give you new Monstera babies if you put them in water.

Instead, cut a healthy Monstera stem/leaf below a node. If it has an aerial root attached to it, you can trim the node to about 2 - 3 inches to fit into your propagation jar or station. The aerial root will most likely black and rot, you can wash the outer black layer off and inside of it is a thin Monstera root. 

Photo source: HousePlantJournal