No, the Kobo Arc doesn't have an HDMI port and odds are that any USB adapter (I'm assuming they exist - don't know from my own certain knowledge) won't work as they've deliberately crippled the USB port so that it can only be used for charging, and for data transfer ONLY when hooked up to a computer.
Yes, the books you buy through whichever app (Kobo's is built in, but you can add the Kindle app no problem) will be downloaded to the Arc as long as you don't choose the archiving options in the app (which would leave it on the server).
Dunno about comparisons to the Kindle Fire in sunlight. As long as you stay in the shade, it's fine to read on. In direct sunlight, I wouldn't bother (on any tablet, even the highest end ones).
No idea how the various markets handle selling you movies, whether they give you a file or just stream. If you do have a file (say, copied from your computer), then of course you can watch it where ever you want. (might not be advisable, though, to try and watch a movie anywhere you don't have access to a wall outlet to recharge, but that goes for any tablet)