Shortcut key to open/apply profile in Terminal

To Create a new Window or Tab with a Specific Profile

You can assign keyboard shortcuts to the items in the Shell > New Window and New Tab submenus using System Preferences > Keyboard > Shortcuts > App Shortcuts to add shortcuts to Terminal.

For example, if you add

Homebrew ⌥⌘9

to the preferences, typing ⌥-⌘-9 will open a new terminal window with the Homebrew profile.

If you want to assign keys for creating tabs, you'll need to explicitly refer to the New Tab submenu using “->”. For example, I have this in my preferences:

Shell->New Tab->Homebrew ⌥⇧⌘9

Typing ⌥-⇧-⌘-9 will open a new tab with the Homebrew profile.

To Apply a Profile to the Current Terminal

You can change the profile of the current terminal using the Inspector window. You can use Shell > Show/Hide Inspector ⌘I to show or hide the Inspector. In the Inspector’s Settings tab you can use the keyboard to select a new profile—you can use the up/down arrow keys or type the name of a profile.

I didn't find a way to launch a profile from Terminal preferences, but I know a workaround.

All the available profiles are shown in the Shell menu.

You can move focus to the menu bar by assigning it a keyboard shortcut in System Preferences:

By default, this highlights the  menu but you can just press the right arrow key twice, down to New Window, right to open the sub menu, then down until you get the right one and press enter.

A new Terminal window with the specified profile will be launched and you won't need to use the mouse/trackpad.