The Merge tool will project your data on the fly, if each input is properly defined.
The output will be either the projection of the first input or
whatever the Output Coordinate System geoprocessing environment is set to.
While it is possible to 'get away' with merging data in adjacent UTM zones
(usually for features straddling the zone division
or features far to the north or south that physically fall into two or more zones)
(the zone edges overlap progressivly as one moves away from the equator)
It is indeed unlikely that a multi-UTM-zone merge would be a good idea.
Similarly, geographic (simple cylindrical) may not be ideal either;
it all depends on what you hope or expect to do with the data post-merge.
Some thoughtful consideration given to a projection choice at this stage
can accrue valuable advantages, and is well-advised.