2005 Mazda6 tranny ****

1. some mazdas had a PCM/TCM update to resolve 1st to 2nd shift shock. the only one who can update the is the dealer. maybe covered under 5 year 100,000kms powertrain warranty.

2, have you ever had the transmission fluid replaced? it could be dirty, burnt, etc.
if it was replaced, maybe it was replace with fluid other than M5 fluid. different fluids have different properties (thickness etc) which could cause serious problems if left.

I would recommend taking it to the dealer for the update and a transmission flush (not a drain and fill) using a flush machine.

It sounds like you're transmission is going out. Try JDMengines.com for a low-mileage replacement transmission. Also, revving it at high RPM's and then shifting it to second will wear out a transmission in no time.