You would have a better chance of people answering if you keep the questions simple rather than asking a comparison between so many words at a time. Also, such a question can easily be answered by using a simple dictionary on the Internet. I will however try to answer you anyway.
*** They all have various meanings. The common meaning being :
to pull or haul (a car, barge, trailer, etc.) by a rope, chain, or other device: e.g. The car was towed to the service station.
They also have some meanings that are different from each other :
*** TOW (Noun) : something, as a boat or truck, that tows.
*** TUG : to pull at with force, vigor, or effort.
*** HEAVE :
1. to raise or lift with effort or force; hoist: to heave a heavy ax.
2. to throw, especially to lift and throw with effort, force, or violence: to heave an anchor overboard; to heave a stone through a window.
3. to utter laboriously or painfully: to heave a sigh.
4. to cause to rise and fall with or as with a swelling motion: to heave one's chest.
*** HAUL :
1. to cart or transport; carry: He hauled freight.
2. to cause to descend; lower (often followed by down ): to haul down the flag.
3. to arrest or bring before a magistrate or other authority
*** DRAG :
1. to search with a drag, grapnel, or the like
2. to level and smooth (land) with a drag or harrow.
3. to introduce; inject; insert:
4. to protract (something) or pass (time) tediously or painfully (often followed by out or on ): They dragged the discussion out for three hours.
to introduce or interject in an inappropriate or irrelevant manner: to lug personalities into a discussion of philosophy.
1. to bring, take, or pull out, as from a receptacle or source: to draw water from a well.
2. to bring toward oneself or itself, as by inherent force or influence; attract: The concert drew a large audience.
3. to sketch (someone or something) in lines or words; delineate; depict: to draw a vase with charcoal; to draw the comedy's characters with skill.
4. to compose or create (a picture) in lines.
1. to pull off or out from the place of growth, as fruit, flowers, feathers, etc.: to pluck feathers from a chicken.
2. to give a pull at; grasp: to pluck someone's sleeve.
3. to pull or move by force (often followed by away, off, or out ).
4. to remove the feathers, hair, etc., from by pulling: to pluck a chicken.