The switch is in world 2-1, before the armour spider, you're talking about world 2-2. Those carts aren't a problem, you can just sprint past them and you won't get hit, there's only 3 of them anyway and I found that I had enough stamina to get down that path anyway. I'm sure you can handle it, but, ya know.
The switch for the lift is at the top of the lift itself. If you're at 2-1 and you've gone through that door next to the tramp, the lift is idle and it blocks you from getting on because it's around chest height. If you go through world 2-1, up to the bit where there's some solid miners with flame pick-axes (also there's about 3 of these ogre things that lob rocks at you, similar to the ones that lob it at you in the beginning of the world, except the ones in question here are adjacent to you, same height).
You must've defeated these enemies before and taken a left through the small fog doorway without pressing the lever, which can be found past these rock throwing enemies, and if you go forward continually you'll come to the cliff which looks out over the very start of the level. If you follow the cliff to the right, you can use the switch. It's a pretty big shortcut.
Basically this lift gives you access to this top level , (where the switch is) where you can get to the armour spider area pretty quickly. The middle level - which is the start of the level. If you wait to go the whole way to the bottom you'll find Blacksmith Ed, the forge-monkey.
Will post video if that doesn't make sense.
Edit: what may be confusing you is this tramp follows you round the stages in world 2, so he may not be where I say he is. Basically the door to the lift (that is stationery in your case is EASILY seen if you spawn at the first archstone in stonefang tunnel (world 2), go up the stairs, and at the top of the stairs when facing directly forward, the door is slightly to your right. You can't open it from here, but you should know that if you're here, the lift is above you somewhere in the level. Not sure if you can see the top ledge from the start of this world.