(1) Answered by @Fuellemann.
(2) Answered by @garageshop. Hey wait, you cheated, you answered your own question!
(3) Likely this is some CSS within that 3rd-party stack that is interacting with the Alert stack and making it not visible. Is it visible in Preview Mode? When Published? If you wish to email me a few things I can look at it further, but I think this one just comes down to the stack you're placing it into has some similar code that is hiding it in Edit Mode.
(4) No, that isn't an option of the Modal stack.