AM I THE DREADED MARE'S TAIL?

Looks very much like it.

Yes.

Don't bother digging it out, the roots can be HUNDREDS of feet long.

Kill it with a glycophosphate weedkiller, or one that kills from the leaves to the roots...it's no use killing off the green bits alone.
Before spraying, "Clap" the weed gently between two pieces of wood. This damages the waxy coating which covers all the green - the coating prevents the weedkiller getting in.
Good luck. It took me two years to rid my 100 foot garden of it. I kept a hand held sprayer at the ready on a daily basis. Evey time one popped out, it got it!