I think what I'm missing with the springs is that you have to look at the second one as the first one is passing overhead, right? Because I started moving when the spring was the furthest away and it still hit me.
It's easy. First two loops, you get to the top of the stair, move left slightly. Next one that comes, move as soon as it bounces in front of you, you'll make it up the ladder just in time. After that, when the thing goes faster, start the same process. However, now you have to do it slightly different. When it bounces up to you, start to walk, but instead of going up the ladder, go forward to the point where it's highest at its arc. Turn right. Wait for the next one, and as soon as it is bouncing over you, move for the ladder, you'll make it every time. Ironically, the first level is always the hardest, the rest of the game is pretty cake.
Edit: I also know this works in later loops. When I first started to play to get higher scores and would die early on, I just kept going to get a feel for the later loops. It all comes down to stupid mistakes for me, otherwise I have all the rhythm and patterns down pretty good. I still was able to use the hammer on upper loops on both levels, you just have to anticipate when the straight and diagonal barrel are going to fall on the first level. I don't even mess with the hammers on the last level unless I know I can kill one of the flame guys, otherwise I use a straightforward left right pattern to get to the top, adjusting as needed and making sure to collect all extra items unless circumstances make it impossible.