There is, indeed, a "Power Off Timer" on the P800, Ian. Mine is set to "off." There also is a "Sleep Timer." Mine is set to "15 minutes."
Neither is an issue. If the printer powers off, it's no different than if you had powered it down manually. Likewise, I'm guessing the sleep timer simply parks the heads, while the printer remains in a ready-to-print mode.
As far as the start-up, as far as I'm able to tell it's no different than the 3800/3880. The same noises. The same amount of time. It's definitely not the explicit cleaning cycle that you can run manually - and which takes considerably more time.
I've printed with the P800 quite a lot since acquiring it on 7/1. I have no sense that ink usage is in any way different than the 3880. I highly recommend it (and given the choice between a new P800 or a new 3880, I would choose the P800).