Your Mac should certainly be able to boot 64 bit. Just to test, restart and hold down the 6 and 4 keys. Then check the System Profiler to see if the 64-bit Kernel and Extensions says "Yes".
Yet, when I look in my Activity Monitor, I see Intel (64-bit) under "Kind" running for almost all the processes except for a couple.
That's normal. The kernel can boot 32 or 64 bit. But all 64 bit software, system or third party, can still run 64 bit under a 32 bit kernel.
Also, does enabling 64-bit mode have any effect on Rosetta?
Is there any disadvantage to running in 64-bit mode (ie: will certain things not work)?
Not really. The only real issue is drivers that must load during startup. If they're 32 bit only, they won't load if you start up to a 64 bit kernel. You would then be unable to use the device the driver is for.
Here's the configuration file approach for always booting into 64-bit. This is the best way to make your Mac always start up 64 bit as the only thing necessary is to alter one line of a configuration file. You can do that by opening the Terminal and entering (or copy the next line and paste in into Terminal):
sudo defaults write /Library/Preferences/SystemConfiguration/com.apple.Boot 'Kernel Flags' 'arch=x86_64'
To return to 32 bit mode, you would repeat the command but enter an empty string, which would just be the single quotes (where arch=x86_64 is in the above command) with nothing in between.
sudo defaults write /Library/Preferences/SystemConfiguration/com.apple.Boot 'Kernel Flags' ''
The forums wrap these commands to fit the box, but if you copy/paste it, they will be one line.