I copied all of my Steam Games onto my hard drive, currently re-installing them, can I recover my saved data?

I have an external hard drive and I've decided to use it to play all my Steam/Origin games. To do this however, I copied all of my game files onto the external hard drive and deleted them off my laptop. I then created a new Steam library. I've been able to get a few games working on it, however, I'm starting from scratch and I was wondering if I'd be able to copy my old game files over and continue from where I left off. I know some games save files in different ways. Borderlands 2 for example, I don't really want to start from scratch with it (if I must, then I must). Any suggestions as to how I could do this?

Also, I've been having trouble with some games that I've put on my external hard drive. I would download them off Steam and everything and then I would play, no problem at all, but then when I unplug my external hard drive and plug it back into my laptop, the game is unable to start and Steam tells me that I must install it again. I would even search through the folders to find the execution file in my Steam library on my external hard drive, but it still won't work. Any suggestions for that as well?