Checking for player movement

You can use the Running event to do this task for you and change a boolean every time it triggers.

Here's some code for you to inspect...

local humanoid  = game.Players.LocalPlayer.Character.Humanoid
local isRunning = false

humanoid.Running:connect(function(speed)
    isRunning = speed ~= 0
end)

Yes, you can. The humanoid's Running event can be used:

local Player = game.Players.LocalPlayer
local Character = Player.Character
local Humanoid = Character.Humanoid

Humanoid.Running:connect(function(Walkspeed)
print("Player has started walking! The walk speed is: "..Walkspeed.."!")
end)

Use the Humanoid.Running event.

game.Players.LocalPlayer.Character.Humanoid.Running:connect(function()
    print("Player is walking")
end)

Humanoids have a Running even which is fired whenever the speed of the character changes. Use that and some logic, and you have a way to tell if the player is walking or not.

local Walking = false

game.Players.LocalPlayer.Character:WaitForChild("Humanoid").Running:connect(function(Speed)
    Walking = (Speed > 0)
end)