Yes... I agree there...When you sack a coach in October (like they did with Rodgers) you have to get a new long-term solution immediately, so I am quite sure they were in contact with Klopp before sacking Rodgers and I agree that could have made that decision easier.
Some kind of "Hey man, what's up?" after he left Dortmund (or similar talk with his agent) is still quite possible (with the probable answer like "nothing for now, just taking my time off work, you got a nice club there I like it"). But I don't believe there was much more contact than this at the time, Rodgers had a contract until 2018 at that point, so I don't think that there was something planned like "Klopp will take over after a sabbatical for season 16/17".
Not sure, but isn't the rumors on managers waiting for a curtin job quite common these days. I'm not comfortable with this, and I guess our history with managers is a reson to that.
So many clubs do change managers easily, to easy if you ask me. For me it looks like the contact between managers agents, and clubs is quite frequent and ongoing over years.