How do I get a part to drop on a location where my mouse is pointing at?

Whenever you want to know where the mouse is pointing, just use Mouse.Hit.p

Also, don't use deprecated events like mouse.KeyDown. Those shouldn't even still exist, they're just around for legacy compatibility.

local player = game.Players.LocalPlayer
local ContextActionService = game:GetService("ContextActionService")

repeat wait() until player.Character.Humanoid

local humanoid = player.Character.Humanoid
local mouse = player:GetMouse()


local anim = Instance.new("Animation")
anim.AnimationId = "http://www.roblox.com/asset/?id=333356213"


local function createNewPart(Position)
    -- Creates new part @Position
    local newPart = Instance.new("Part", game.Workspace)
    newPart.CanCollide = false
    newPart.BottomSurface= "Smooth"
    newPart.TopSurface= "Smooth"
    newPart.Material = "Slate"
    newPart.Position = Position
    newPart.Shape = "Ball"
    newPart.Size = Vector3.new(4,4,4)
    newPart.BrickColor = BrickColor.new("Dark stone grey")
    newPart.Transparency = 0
end

local function onKeyPress(actionName, userInputState, inputObject) -- The new way we are detecting input
    if actionName == "V" then
        if userInputState == Enum.UserInputState.Begin then
            -- When they hit v, even if they didn't let go
            local newPart = createNewPart(mouse.Hit.p)

            humanoid:LoadAnimation(anim):Play() -- Play Animation

            spawn(function() -- Run on a separate thread
            game:GetService("Chat"):Chat(game.Players.LocalPlayer.Character.Head,"text","Red")
            wait(1)
            game:GetService("Chat"):Chat(game.Players.LocalPlayer.Character.Head,"text","Red")
            wait(1)
            game:GetService("Chat"):Chat(game.Players.LocalPlayer.Character.Head,"text","Red")
            wait(8)
            newPart:Destroy() -- Still 10 seconds before this gets destroyed
            end)

        end
    end
end

ContextActionService:BindActionToInputTypes("V", onKeyPress, false, Enum.KeyCode.V) -- Whenever a Player hits V, connect