Can a laptop be switched on all day when plugged into the wall and used as if it were 'desktop' machine -- how does this affect the longevity of the battery (I've heard mixed reports on this)?
Li-Ion batteries have their lifetime defined in terms of "recharge-cycles" (around 500-700 these days). That means regardless of how little or much the batterie is depleted, recharging it means a decrease of 1. The capacity then dignificantly drops. Depending on the powermanagement the laptop has, having it constantly plugged in -can- greatly reduce the life-time of the battery (because it gets recharged from 99 to 100% from time to time), but I guess (do not know) most modern Laptops shouldn't suffer from this.
Could I run say two large widescreen monitors off a laptop -- or would I have to stick to a powerful desktop for that?
No problem there. Laptops usually come with nice gracas that will do most of the work.
There should be at least one port to plug in a second monitor. Even my 6 year old got it. Larger ones may even come with 2 of them. (Doesn't make too much sense if it comes with a large screen already)
proz:
Upgraded in early 2009 to an i7 with 24 Gb and a dual 4870. It comes close to 250 Gflops and can run 8 parallel processes, but unless you're rendering videos all day long or doing excessive FEM/CFD calculations, the i5, which is a lot cheaper and yet almost as powerful might be a good choice. Otherwise I think you'll be fine off getting a standard quad-core, which are ridiculously cheap these days and still powerful.
complete/self-built:
You can get a complete and up-to-date PC in various stores for a few hundred bucks. Building one for yourself might cost a few extra-bucks, but will give you the nice feeling of having it built for yourself and with care.
breaking the daily routine:
There is one big advantage coming with a Laptop: you don't have to always sit at the same spot when working (which I know can be annoying). You can work in bed, in your living room, on the train, wherever.
another hint:
don't tear apart your old comp just for the harddrive and the casing! Built a new one from scratch and have it connected with the old one in a local network + using a kebord&mouse switch, so while one is busy, you can work with the other just by the flick of a button (usually a double tap on CTRL). You can then join the work on your main (new) computer using the shared drive.
lifetime/warranty:
It seems all modern Laptops have a built-in selfdestruction which is activated exactly after 2 years.
The most common problem causing the failure of a Laptop is high temperatures destroying the harddrive (flash drives shouldn't suffer from this, but are still rather small+expensive). It is wise to turn on the fan constantly at low rpm in the BIOS.
my tip:
- built a cheap (400 bucks max) Desktop PC from scratch.
- get a used Laptop or larger netbook for ~100 pounds, with a batterie that still lasts for an hour or so. You really should be able to show some of your work to your customers directly. Don't let go pen&papaer though.
all of these were specific needs for me.. it also does well on heat throwing it out one side and the base barely get's hot except for one small area. Another thing I hate about it is the glossy screen.. reflections prevent me from seeing the screen very well... but it's something I didn't know about until I got it. 
