Is there a keyboard shortcut to bring up Finder? Is there a default keyboard shortcut to start a new Finder window?
If not, is there a way to assign one?

I tried System Preferences > Keyboard > Shortcuts > App Shortcuts and created a new item for Finder, and assigned shift-command ⌘-F, among other things, but nothing worked.
It's actually easy. On Yosemite, just press option ⌥+command ⌘+spacebar. That will open a new Smart Finder window. You then can navigate from there. It works on a system level no matter what application you're running. 

No need for scripts or complicated setups. 

Turn this option on in System Preferences > keyboard > shortcuts > spotlight > "show finder search window" in recent versions. This can be mapped to other keys too, but it can cause conflicts in other apps that might be using your desired shortcut.