Mouse Hover detection/Key hit?

Yes you can, and it's actually quite easy.

There is a property of mouse called Target that is equal to the object in the Workspace that the mouse is currently pointed at. Knowing this, we can write a script that checks to see what the mouse is pointing at each time the mouse is moved. If it is pointed at the door, we will call a function to display the billboard GUI. However, we will have to remove the billboard GUI each time the mouse is move or else they will never disappear.

You can go about this in two ways, either clone and destroy the GUI or simple set the visible property of the GUI objects depending on if it should be shown or not. These examples should be run in local script

####Example 1:

local player = game.Players.LocalPlayer
local mouse = player:GetMouse()

local door = game.Workspace.Door
local gui = LOCATION_OF_BILLBOARD_GUI

mouse.Move:connect(function)
	if mouse.Target and mouse.Target == door then
		gui.Frame.Visible = true
	else
		gui.Frame.Visible = false
	end
end)

####Example 2:

local player = game.Players.LocalPlayer
local mouse = player:GetMouse()

local door = game.Workspace.Door
local gui = LOCATION_OF_BILLBOARD_GUI
local copy

mouse.Move:connect(function)
	if copy then
		copy:Destory()
	end
	if mouse.Target and mouse.Target == door then
		copy = gui:Clone()
		copy.Parent = DESIRED_LOCATION
	end
end)