okay...1 if you're tearing up 60 grit sandpaper then your stripper needs another go around...it probably just took off the paint down to the varnish, shellac, polyurethane whatever was protecting the original stain. 2 I wouldn't try to put stain on something that hasn't been taken down..it has to absorb into the wood to work...if there's still some of the varnish on it then it won't adhere properly and when you go to polyurethane it your stain will just come off and it'll be all 'blotchy' and finally...if you're wanting an 'old' (i.e. dark) look don't worry about taking the stain all the way off..I use Varathane Espresso. If you're wanting it lighter then yes..you'll need to clean up the previous stain.
p.s. Check the stripper you mentioned to make sure it will remove varnishes and shellacs..might try just plain old acetone but if you're tearing up sandpaper again it's still not 'clean'