That thread has a wealth of info.
As for the springs, you need to run to the edge of the first elevator and wait there. Trial and error will net you the "safe spot" where the heel of the springs will graze your nose, but not kill Mario.
If you watch the springs carefully, you will see that they vary in their arcs, some bounce farther forward, some hang back. Watch where they land in relation to DK's foot. What you are looking for, is one that bounces farther forward. When that one passes over Mario, you can get a head start running to the right to catch the ladder. The split second of head start lets you avoid all other springs EXCEPT if the very next spring is again one that bounces farther forward. So you have to start running to get your head start, and watch the next spring. If its a forward one, retreat back to your safe spot, and wait for another forward spring. If its not a forward one, power ahead as quick as you can to make it.
Hope that helps.
Edit: This video shows the basic idea, the details about the spring locations and the timing you will have to work out through trial and error though.