According to this page, the 3810 didn’t have the auto power saving feature, but the 3820 does. I think pressing home is acceptable for saving a bit of power, but if you disagree, that page shows how to turn it off.
A dancing Roku logo is something I’ve only seen once or twice in 6 years – that means there was a full power down of the device. But if the new one is plugged in in the same way (presumably into the TV’s USB port which switches off when the TV is off) then I don’t know why you wouldn’t have to wait for a full boot up on that one as well. By the way, some TV USB ports don’t supply quite enough power which can lead to various strange behaviors.
I see no Samsung app in the Roku channel store, so anything you see from Samsung is your TV, not the Roku. When my Samsung is switched to the Roku input, and I haven’t woken up the Roku, it just displays “no signal”. But yours may have a different feature set.
My Samsung also always turns on to the last-used port. I don’t use HDMI-CEC and switch it between the various inputs myself.