Do I need to have system administrator permissions to install from the Mac App Store? Where does the Mac App Store install its applications to? Is this a system-wide location, or somewhere inside my Home directory? Is it possible to use the App Store on machines where I am not a System Administrator (just like I can currently install almost all applications I manually download by putting them in ~/Applications instead of /Applications)?
Yes.

Apps are installed to the usual /Applications folder. Non-admin users can browse the store, but they will need an admin username/password to install apps. Source: Help menu of App Store/Purchase Applications/Buy, download, and install applications/To buy, download, and install an application:/Step 4.