I am curious how are situations with extra player in ice-hockey handled statistically. For example, team A is loosing 2:3 and there is one last minute of the match. They switch goalie for an extra player so they have empty net and the situation is 6v5. If they score a goal, will it be an even strength goal? And what if team A was playing short-handed, they had penalty and it was 4v5 situation but after switching player for goalie it would be 5v5. Will it still be a power-play goal? Would it be even strength goal? Or how would be the situation described, just 4v5 or 5v5 (maybe 4+1v5)? And what about 6v4 etc.? Thanks for answers!
Yes, they are recorded as a normal, even-strength goal. See for example TOR vs MON, Oct 5, 2019, Hockey Ref box score. Austin Matthews scored the tying goal with the goalie, Michael Hutchinson, pulled; you can only really tell from the box score as Hutchinson has only 64:52 TOI. Matthews is credited with 2 Even Strength goals. Even +/- it clearly counts for; add up TOR's +/- and you get a net +1, while adding up MON's +/- you get a net +0 (which makes sense, as both sides had one goal that is excluded, one PPG and one penalty shot, so both sides scored 4 even strength goals - including the one scored with the extra attacker).
Another game from the same season, CBJ vs BUF, 10/7/2019, shows a powerplay goal in that situation - there it is a 6 on 4 goal, if I'm reading it right. Similarly, it's not really identifiable as anything beyond a normal PP goal, except for the goalie's TOI again being less than that of the other goalie.
In the case it was a 4-on-5 with a pulled goalie, it would be a short handed goal, similarly. Removing the goalie, at the moment, doesn't change the statistics, other than to have one more player receive a +/- result.