Move it
Drag the window by any empty part of its background. There is no title bar, but the whole window is draggable.
Resize it
Drag any edge or corner. The window remembers its size.
Keep it open
By default Mate lives in the menu bar and closes when you click outside it, which is what you want for quick look-ups. If you prefer a window that stays put, open Settings › General › Show Mate in and choose Dock. Mate then behaves like a regular app: a normal window you can move, minimize, put next to your document, and switch to with Cmd+Tab. Choose Menu Bar to go back.
Older versions had a pin button; it's been replaced by this setting.
Open it where you are
Press Option+Shift+T to open Mate from any app. Change the shortcut in Settings › Shortcuts. See Keyboard shortcuts in Mate.