Basically, yes. The Remote Lite II, like the Epoque, Sea&Sea, Inon etc
flashguns have a built-in slave trigger which will fire the flashgun when
another flashgun, not conected to it physically, is detected as having gone
off. The flash of light from your digital camera's built-in flashgun may not
be sufficient to illuminate the subject, but it should be sufficient to
trigger an external flashgun in this manner. This method of flashgun trigger
is well known in underwater photography circles, but is only now catching on
for land photography. Of course land photography does not have the
constraints of making everything work when bathed in an electrical conductor
(sea water) nor are you constrained in physical size (having to fit
everything into a housing). As more digital cameras are beibng released onto
the market with neither a hotshoe nor a cord connection for synchronising
with an extrernal flsdhgun (should one be desired) this light activated
triggering of a slave flashgun is gaining popularity. Sunpak and Vivitar
make smal, economical flash units which, like land cameras, may work well
when placed in suitable underwater housings. I stress that I have yet to try
this, but cannot see why it should not work.

No reason why you can't mount a flashgun housing onto a camera bracket, in
the same way as you would on land.

Also possible

Precisely what you have understood it to be!

If you have very powerful lights you may deform the plastic with disastrous
consequences. There are of course purpose-made underwater lights for video
cameras, but the price is astronomical. As to the switching on and off, that
will require some engineering well beyond my capabilities. My idea for a
land flashgun inside a transparent waterproof box was to have nothing to
control. I would switch the flashgun on before closing the lid, and only
switch it off at the end of the dive once ashore / aboard. Of course land
flashguns do not consume much power when left on and not flashing, and there
may be room (depending on the box chosen of course) to mount (for example)
three pairs of AA cells to power the flashgun with, rather than the one pair
it normally may hold. After all, there may be spare room in the box and
changing batteries half-way through a dive is not an option! Rechargeables
are inexpensive enough and can be rendered fully charged before the next
day's diving.

Now if you're placing a video camera inside an otterbox, another poster has
alluded to the optically imperfect nature of the box material. For a
flashgun or other light this would not be an issue. It would be as if using
a diffuser at all times, that's all.

These land flashguns from Sunpak (Remote Lite II) and Vivitar (DF120) sell
in the UK for around ?45 (E65, $85) and ?30 (E44, $57) respectively.
Chucking them in a box which costs ?20 (E29, $38) seems a better deal than
the least expensive of underwater flashguns I've come across at around ?250
(E362, $475). You also get some degree of auto flash exposure to boot.

Remember not to take my word as gospel - I have merely used postings here to
sound out ideas I've had. Do not assume my ideas are bound to work, I assure
you they often don't! If you choose to embark on any projects based on these
thoughts, the liability is yours. In this litigation conscious age such
statements appear necessary, unfortuantely.

Ken