Is it bad practice to connect with your interviewers from LinkedIn from a company you were rejected from?
Use of LinkedIn, and similar services that connect professionals, varies a lot between individuals.
Recruiters on LinkedIn will often try to make "cold" connections, where they don't know the person, and are hoping that to use the connection to place them in a sales funnel (either as a candidate or employer).
Some people will be willing to make connections after a meeting, especially if the meeting sets up a new working relationship.
However, most people that I know tend to make connections as follows:
Valued colleagues from current or previous jobs, close friends and family members are actively searched and added
Employees at current company are added as and when recommendations come up, and quite often a quick review and bulk addition is done by people leaving a company on good terms
Of course, this could be selection bias, and I am only connected to other people like me. Searching a few articles about this, this sort of approach does seem to be the norm, although I could not find any solid statistics. My general impression is that adding new connections on LinkedIn is done conservatively by majority of users.
After interviewing someone, I would not personally feel that I had made a long-term connection. I cannot speak for you, or your interviewer, but I can say that I would be surprised to get a connection request from someone whose only connection to me is that I had recently interviewed them.
Unless the interviewer had already suggested that they would like to maintain contact, or there is clear evidence that this would be normal for them, then I would not personally bother. The value of the connection is not likely to be high for you either way, and there is a possibility that your request is viewed like a recruiter's cold connection.
Having said that, LinkedIn does make rejecting connections simple and guilt-free, so you don't have much to lose either. Don't worry if you have sent out lots of requests, and don't read much into it if you get lots of rejections - it will just be that your definition of "we have a connection" has a lower bar than the recipient's.
No, I don't think so there is anything wrong with that. I'd say it was a smart move, especially if you connected well with them during the interview. You might not have been right for that specific role, but you might be right for a role they need to hire for in the future, perhaps at another company.
Remember that hiring decisions tend to be a group consensus thing. If you aren't hired, it doesn't mean that everyone who interviewed you thought you weren't right for the role. Sometimes it only takes one person to have reservations for the hiring company to keep looking.