Good to hear that you found someone to replace the battery (or the hard drive)! Is he from Wisconsin, by any chance? The reason I ask is that there is a guy near me that replaces the drives too (don't recall if he does batteries too). These folks are few and far between, because it isn't as simple as just throwing in another drive.
I don't recall if the info about the 605 needing to be left turned on while charging is in the manual or not. But I know that it has been mentioned in these forums a zillion times. By trying to charge the 605 with it turned off somehow bypasses the circuitry that determines when the battery is finished charging. Then it can overcharge, damaging the battery. Also, don't leave the battery charging 24/7. Even though it theoretically stops charging once full, there is still a small trickle charge keeping it topped up. This small extra charge being applied over long periods of time can also damage the battery. I have read here of rare instances where this constant small charge being applied over long periods has caused the battery to actually physically swell in size and put pressure on the hard drive, which then doesn't allow the 605 to function. And if you don't plan on using your 605 for extended periods of time (say a month or two), it is best to store it with about a 40% (or so) charge, rather than fully charged. This will extend the overall life of the battery (according to the Battery University website).
Anyway, hope you enjoy your new toy!