Thanks so much for sharing your experience with us…it helps in knowing what issues exist in real life.
1) I would hope that the grounded chassis would keep this from happening much, but power transformers do emmit a lot of flux. If you want to keep it on your amp, maybe try to keep the pedal on the side of your amp away from the power transformer (the side where the guitar input is). Also, mind how your input and output cables are laying on the amp top. They can pick up hum too.
2) Yes it can add emphasise some higher frequencies, but you can “tune” it with the high and low trim pots to get more of a natural sound.
3) Amp sag should just happen when you hit the gutiar hard, whereas the sag from the tremolo LFO will happen in steady time. One idea is to set the tremolo LFO faster (like 2:1 or faster ratio) so that when you hit the amp hard on the down beat the LFO can still be hard and the two phenomena don’t have as much effect on each other…just an idea.
4) The battery should be fine for short stints. I estimate around 6 or 7 hours of use with a typical 9 volt battery.
Your 9v/300mA power supply is fine…the pedal uses around 50mA, so anything more than double that should be fine.
Thanks agian for your comments.
-Brach