Well, there is a lot of advice above. So I will concentrate on only one move,

(C6). I think it is interesting because the play epitomizes what we do not want to do in a tight situation.
Let's face it, this kind of thing happens to us all the time. We are cruising to a great result when suddenly our opponent rocks us back on our heels with an unexpected (because we're winning so we don't really check, right?) cut that exposes weaknesses all over the place. What do we do? We have to buckle down and make the best of it. This means carefully considering what the situation is and what resources we have. This does not seem to be what happened in the 19 seconds that Black pondered before playing C6.
We are cut twice by

and

below. The play at C6 basically retreats from both. There is no way that White should get away with that. Consider:
* Locally in the corner, White has some very nasty looking aji if Black plays "a" (ko at best?).
* On the bottom side, the White stone at K4 will be isolated between two Black positions if Black captures with "b". On the other hand, if White saves the ladder stone, the K4 stone will become too tight an extension, so either way Black should not fear the result on the bottom.
* On the left side the White stone at C10 squeezes the Black stones severely if they are cut off. On the other hand, if the Black position is not cut, the C10 stone may end up too close to a strong Black wall and be an inviting object of attack. At the same time, if the Black stones are cut off but live, it is likely sente against the lower left corner due to the bad aji there. So here too how worried should Black be?
In the game

meant that White had broken the ladder on

without any damage to

. I think if I had been White I would have only pushed up once at

below and then turned at

, trying to settle the bottom left without going back to capture at G2. Pushing again as in the game eliminates the aji of pulling

out with "b" or playing atari with "c". Black would have been uncomfortable if White had avoided erasing this.
I think that

below is probably the most reasonable play in this position. What is White going to do now? If White pulls out the cutting stone with

,

activates the threats in the corner. Meanwhile on the outside White has difficulty in continuing. There is no good way to strengthen/develop the cutting stones without strengthening Black at the same time.
Likewise,

performs much better than C6 if White pulls out the ladder stone with

below. Unlike the game the cutting stone is captured after

and White has no further forcing plays here. So White cuts at

. However, Black has the ability to squeeze White with

if the game later warrants.
If White tries turning in this situation, there is no aji above so Black can easily play the triple hane with

, intending to squeeze or perhaps turn against the K4 stone.