Fine work!
Mahogany has a very attractive dark reddish-brown colour but its main advantage is being strong and close-grained.
I'd think there is any hard and fast rule about spacing the lagging bands, but start with one round the top and the bottom of the wood (the lower being the foundation-ring level unless you intend taking it full-depth for better appearance).
Then I'd have one each at the thrids points but that might be very easy here due to all the fittings, fire-hole etc. I'd be inclined to experiment to see what is most pleasing as well as mechanically sound. You might find a single central band would both look and work well.
The method you describe for joining the ends of the bands is as common and as good as any. A solid but neat block on each end is certainly better than simply folding the strip up, which the screw would try to straighten out so look poor.
Under the wood? You won't have much room but a thin insulating layer will certainly do no harm. The total thickness of blanket and wood is largely dictated by the fittings and their bushes, allowing you to remove a fitting if necessary without having to disturb the woodwork.
It may well be better not to glue the boards together. I'd be tempted to machine a very shallow rebate, fractionally under the band thickness, on each end of each board to engage the rings, to stop the boards moving vertically.
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On boiler testing, as this is copper it should be inspected first un-clad, but not need the cladding removing in future. I'm not sure but I don't think setting the safety-valves hydraulically works all that well. I think you'd still need re-adjust them during the steam-test, so they match the red line on the gauge.