I am no expert but I have always had a very difficult time getting all of the paint off of the wood, it seems to penetrate the grain of the wood so deeply it is ( for me and my limited skills) nearly impossible to get off all of the paint. Sometimes you can't go from paint to stain. You might want to try stripping a spot to see how clean you can get the wood, and then try your stain.
It is a huge undertaking, when I refinished my meadowbrook I did not take it all apart. I took the slats off of the floor, make sure you number them, I numbered every one on the bottom left so I knew top from bottom and right from left. I did not take anything else apart.
For your first refinish job, unless you are very handy or have some help, I suggest, don't take the cart all apart. They are not easy to get back together, bolts get stripped and nuts won't come off, screws are stripped. I have sat with an entire sleigh taken apart in my living room for an entire winter, because I needed to replace some wood on the runner, that was a nightmare.