Once a day, roughly, two competing blocks with the same height are mined, which leads to one of them being orphaned.

Was there ever a "natural" case in which the fork lasted for two rounds? By natural I mean a fork that was not created by any bugs such as the one in March 2013 ( https://bitcoin.org/en/alert/2013-03-11-chain-fork ).

What was the longest fork?

A naive back of the envelope estimation says that it should happen once in 100 days, under the assumption that half of the network heard about each of the competing blocks first. Therefore, a more realistic estimation would be once in a year or two.