I can't seem to start a new game. The game loads, just can't start a game. Even verifying game cache did nothing for me.
Why would one need to reinstall the game? Can't we just move Dawnguard out of Data (hide with Bash) and then make sure plugins.txt/use BOSS is done?
[edit] Seems this is related to SKSE. I can't start a new game with SKSE active, sometimes I can but then I can't reload the save.
1. Check if SKSE is installed correctly. The easiest way to do this is to just reinstall. I had problems with my SKSE because files keep spontaneously disappearing..So every now and then I do a fresh install of SKSE to replace the missing files. Last time it happened, I only had 72/345 SKSE script files. So Just try and reinstall it, it might work.
2. Check that your SKSE is up to date. Current version is 1.5.11. To check if it's up to date, right click the SKSE launcher, Properties>Details Tab.
3. Check your mod load order. Sometimes problems are caused because the mod load order is incorrect. BOSS does correct the load order somewhat, but not entirely accurate. So after using BOSS, check the mod load order and make sure everything is where it should be.
I just installed Dawnguard, but when I try to play the game it crashes as soon as I see the logo.

I tried to start it with mods disabled, but it doesn't work either. Then I tried it with Dawnguard enabled, and it works.
What can I do to play with Dawnguard now?
How can I check if the download was 100% complete? Do I have to put the Dawnguard esm. on top of the list?
I didn't uninstall my mods prior to downloading Dawnguard - how can I try to install it again with the mods deinstalled? Is there a way to uninstall Dawnguard and try again?
Any help is appreciated.

As someone else mentioned, if it's crashing before the logo, verify the integrity of your game. If there are files missing, doing this will download and replace them.
If it still doesn't work, then it's likely a mod.
Check that all your mods are up to date, and make sure that they are compatible with Dawnguard. Also check you mod load order, as that could cause problems. If you use SKSE, then check the about 3 answers I gave to Psymon.
EDIT:
Also, reinstalling the game entirely is a LAST RESORT, in my opinion. It's not always a necessary thing to do. Usually, most problems are fixed without having to reinstall.