A couple basic thoughts on puppy training:
Never let the pup out of your sight. If you miss the moment when the oops happens, you're too late for even "No!!!" If you catch him in the moment take him outside. Over and over.
I don't think a pup would "get" the message you're trying to send by picking up the poop and taking it outside. Like Sharron said, lots of praise for going outside.
Regarding the sitting at your feet he's probably just trying to be close to you, wants attention. And maybe he thinks that the leash thing is a game? Is that a possibility?
As far as stairs go, try luring him up them on at a time, with treats. And lots of praise.
How much do you know about your pup's life up to this point? As in how many people, sounds, smells, things has he been exposed to through strategic socialization? There is definitely so much to think about as far as giving them a well rounded experience. Like my girl Zip is iffy on anything resembling live drums and music. She gets really stressed. We've had to leave a great park more than once because of something she's heard that freaks her out, like music coming from the restaurant across the street. Maybe spend some time thinking about the kinds of things you want to do with your pup and the places you want to go and start introducing those things to him. It's definitely a good time to do that.
Good luck!