It should work, as that is why they included it. Though, do you have a DVI capable monitor for at least temporary use in case you run into any problems? If not, that's still fine, as like I said it should work with either hookup.
No, I don't have a spare monitor. I'm replacing my old setup where I was using a standard VGA out, as my TV supports it. I'm monitor-free for several years now I'm getting one of those Shuttle SFF and wanted to get the low profile card. The regular size cards do have VGA out, but I wanted to go for the low profile one..
Yep, that should work great for that. I have a similar card (Sapphire 4650, but full height plus DDR3) and it works great and stays nice and cool. The only thing I am not 100% sure of is the noise. I don't think it is terrible or anything, but I haven't actually checked to see if it is silent or whatever, and for HTPC that can be very important.
Based on some reviews it looks to be fairly silent. My current setup is ultra quiet, but it uses on-board Intel card and has an external, fan-less PSU. The internal fans make virtually no noise, as there's not a lot of heat generated. Pretty cool