Cut:
To cut and run from something is to rapidly depart a situation. It is
another naval expression. In the old days anchor "cables" were made of
rope. If a ship was at anchor and suddenly came under attack the crew
would not attempt to raise the anchor; rather they would cut the rope
and allow the ship to run before the wind.
To
cut no ice means to have no influence. The saying comes from the USA
and actually means what it says, unlike many in this book. The cutting
here is done by ice skates and, if not sharp, will not allow the wearer
to slide easily over the ice. Blunt blades make no impression - they
cut no ice.
To
cut to the quick means to cause deep emotional hurt and has the same
origin as the theological Quick and the Dead. Here "quick" comes from
the old English Cwicu meaning "living" and thus to "cut to the quick"
implies a wound into living flesh.
Cut out: 'To have your work cut out for you' means that there's a large
amount of work to be done. An odd phrase, and surely very difficult for
a non-English speaker to understand and use. The most likely origin is
from dressmaking or carpentry, where pattern parts are literally cut
out so you see what needs to be assembled. Whatever the pieces are, the
mission is clear, and the work to be done is evident; your work has
been cut out for you - get on with it!