Hey Avidaxis,
Yeah, the HH uses alder as its body.
For the other Jag, I use a mustang bridge instead of the original jag bridge.
I've set the height of the bridge to my liking and there's no more buzz and yeah, it stays in tune. The bridge will rock as it is design as such, for the floating tremolo system..The bridge's 'legs' are resting in 'cups'. For me, what i did was to push the bridge upwards/forward toward the bridge pickups, so it will stay in place that way. It is still a nice sounding guitar, but is one, where constant care and maintenance is needed to achieve its capability. just my humble opinion.
I've attached a pic of the bridge, just in case.
thank you!.. myBIKE.. a 58 schwinn which i restored!... i posted the same background (bike) with my bass... same here, i recieved more compliment with the bike rather than my gear! hahahaha....