Re: mistakes made during install
Sounds to me like what you did was fairly reasonable for your first attempt, and pretty sure this will all end well.
I would recommend checking, as you said, that the queen has been released OK, only because from what you say I'm not sure where the cork ended up relative to the exit hole & if she can get out. If the queen is still in the cage when you check, just release her and shut the hive again, don't do anything else. Unless the package is still in there, remove that. Remove the package first, then release the queen and shut the hive.
EDIT - me and Bison kinda posted at the same time so I've repeated a bit what he said as I didn't see his post till after I posted. Agree totally with Bison, good advise.
44 years, been commercial, outfits up to 4000 hives, now 120 hives and 200 nucs as a hobby, selling bees. T (mostly).