Will using conversion on flying press make a Pokemon a Fighting type or a Flying/Fighting type
The Porygon (2, Z, etc.) would turn into a fighting type because Flying Press is a Fighting-type move. This is similar to Hawlucha only gaining one STAB from using it, Normalize making it a Normal-Flying move and Electrify making it an Electric-Flying move.
Sources: This and past experience with Reflect Type.
will using conversion on hidden power make a Pokemon normal or what ever type hidden power is.
It will make the Pokemon a Normal type, therefore making the move fail (the only Pokemon that can learn this are Normal types).