Is there a relation with the noun planet and to be in a plane?
There is no close connection. Planet is from Greek asteres planetai meaning wandering stars (via Latin and French). Plane is from Latin planus meaning "level or flat".

It is possible that both words are derived from the proto-indo-European root *pele-, (flat) However the evidence linking *pele to planetai is weak. Thus there is no more connection between planet and plane than between planet and field (field is also derived from *pele, via the sound change in German from p to f) 

http://www.etymonline.com/index.php?allowed_in_frame=0&amp;search=planet

http://www.etymonline.com/index.php?allowed_in_frame=0&amp;search=plane

http://www.etymonline.com/index.php?term=*pele-%20(2)&amp;allowed_in_frame=0