It won't melt your charger. Bosch intends for you to be able to use a single charger to charge thru the single port on your bike, with both batteries installed. I do it all the time. Also I have a Tern GSD that supports dual battery from the start and I've done that also.
The range boost will double your range if you run a 500Wh battery already. It's an extra 500Wh! My Domane+ HP goes from 25 miles to 50 miles (for Turbo, as an example). If you have a 625Wh battery already, then you'd get more like 80% more range.
The bike kinda learns a bit as you ride it more but also the warmer weather will definitely make your battery happier and provide more range. It's why my old Nissan Leaf got 40 miles in sub-zero temps and about 90 in the Summer.
Should be fine but you'd have to change the rear cassette and maybe also the rear wheel if the freehub isn't compatible. But there have been some lower cost eTap stuff coming out that might work on your existing freehub.
The light requires two little screws so you could maybe zip tie thru those holes to the rear of the aftermarket saddle rails. You could also order the rear light from the previous year's Domane+ but it's not as good...
Yeah it is designed to accept a Bontrager rear rack. Just make sure you use the right screw lengths and spacers for the upper mounts (they should be included). There are some specific instructions about how to properly install it.
Slide the battery up and down along the down tube to make sure it doesn't move TOO much. A little slide is okay but if it's like 3-4mm lengthwise, ask them to adjust the upper or lower battery docking a little tighter...
The Domane+ ships with a narrow-wide ring. But you need to make sure it's indexed on there correctly. There is a small white graphic that shows how the chain should be set on it w/respect to the wide/narrow links.