I didn't keep track of what my time was at checkpoints, but what I will tell you for starters is that the Alpine is easily the fastest car for this. As for everything else:
-Redlining the car at the start is not the fastest you can get the car to start. I got a faster start when the car takes off at about 7000rpm and a way I achieve this is I hold the accelerator before your car even revs, let go right when it reaches its max rev, let it drop to 3000rpm, and then floor it. It's somewhere in that range, and you'll know you'll have done it right if the car doesn't redline like crazy or if it sounds like it's picking up reeeally slowly.
-The next thing, likely the biggest thing that will affect your time is how much you let the car slide. If you've driven this car at least once, you should already know that this guy loses a lot of speed when it slides at deeper angles and accelerating at this time makes the situation worse. Well, all this means is that you're going to avoid drifting as much as possible as a rule of thumb. However, this car also has crummy handling and gripping through some turns just takes too long, so what I do for turns like these is I use low-speed light-angle drifts so the car is oversteering enough to get me turning, but not enough for me to lose a great deal of speed. Learning how to do this I feel is crucial for this game. To reduce the amount of speed loss here, I forcefully countersteer, let off the gas, and reapply the acceleration carefully keeping in mind not to get too much wheelspin so I can get back up to speed sooner (applying this kind of mentality opens your mind about how you can get out of corners sooner). Where did I apply this? IIRC, on the first turn, the right bend leading to the first hairpin, the tight hairpin itself, an the two other hairpins that have the fence running along their inside apex.
-How often you should be braking? On the first right turn, the second right turn if you're traveling in really fast, the next hairpin, the left turn at the t-intersection, the next left-turn hairpin, the following 2 hairpins after that.
-Where can you be cutting? Lots of places, but remember that causing your car to life off the ground isn't going to help, so avoid any potential bumps if you can. Right at the beginning, yes, right where you start. Instead to steering left and right under the toboggan ramp, imagine a straight line that travels through these two small turns and take that. Avoid getting too close to the bridge as there is a bump there. Get on the left side of the road so you can take the following right turn better. You can cut into the thicker snow when you passed that one bump that's sitting in front of it. You don't need to cut anywhere afterwards leading up to the hairpin. After the hairpin, there's that awkward looking right turn. If you put half your car in the snow and turn a little, the impact of that turn will steer your car a bit as well. If you get the hang of this, this will help you turn faster if you don't let it throw the car sideways. The next turn is the intersection, the end of the ski slope. You can cut this without braking, just remember that your car will get sideways from this turn, so don't get into the turn with too much angle already and be prepared to counter-steer to get it out of the drift as soon as you can. Don't cut the next two right-turns like the Pace-note man says, cut the following left over the grass and twigs. After you go through those hairpins, cut through the thick snow that sits just past the bridge, and on the wide right bend, stay close to the inside, and cut through the snow on the last turn. Avoid driving in the deep snow for too long.