Was my new carpet installed correctly?
Nope, your carpets weren't installed correctly. I'm not sure if code dictates that tack strips need to be replaced, but at the minimum, your contractor should have checked if any tacks were poking through or would otherwise pose a needle hazard. Those things hurt.

Shedding seems to be normal. We recently had some carpet installed on our basement stairs, and some corners (and manually-cut straight edges) do have the little plastic strings hanging loose, and several nylon curlicues roaming around. I really wish all installers would properly vacuum their junk up, but most of them don't.

Carpet on the steps does not necessarily have to be a single piece, simply because you can start a new piece (and stop the old strip) where a tread and riser meet. As long as the edges are butted together with no gaps, you should be fine.

Also, simply vacuuming should not lift carpet edge off (anywhere, specifically not stairs!)... that alone should require that they come back and fix it, or pay for a professional to re-do the effort.