Phrases: "beat one's time" and "cut away"

Posted by Sphinx on July 17, 2003

1.(a short passage)
Foot-racing is a popular activity in the United States.Though serious runners may spend months training to compete, other runners and walkers might not train at all.Those not competing to win might run in an effort to beat their own time, or simply to enjoy the fun and exercise.
(what does "beat their own time" mean? (any alternatives?) is "beat" the same as "outdo" here? what does "to outdo one's time mean" then?)

2.(a short passage)
 About 1 o'clock a storm came up suddenly. Before we had time to do anything, we were blinded by snow. After two hours of this, I realized I would have to do something to avoid being frozen to death. From time to time through the mist, and I had made out the outline of a dark buttress just above us; to descend in this wind was out of the question ; our only hope was to scramble up to this buttress, and dig out a platform on which we could put up our tent. We climbed to this place and started to cut away the ice. ....
(can "cut away" be replaced with "cut out"? what does it mean? what about "cut off"? i'm almost confused by these phrases!)