"is" vs. "are"

I have tossed these sentences to and fro and cannot determine a correct solution.

Should the blank in the following sentences contain "is" or "are" (grammatically)?

1) Each winow and each door ____ locked.
2) This is one of the facts that ____ funny.
3) Dice ____ the solution to my random number problem.

In sentence #1, the objects are taken one at a time (each), so the answer should be "is". However, it seems to me that since both are taken one at a time separately, and then joined with "and", the correct solution should be "are". If the sentence was "Each window and door ____ locked." then it would be "is" because the windows and doors are taken one group at a time. However, when I put "each" to both objects, does it change to "are"?

In sentence #2, I think it could be either one. For instance, if one considers the sentence to mean "this is a fact, and it comes from facts which are all funny", then the answer would be "are". However, if it is taken to mean "this is one fact and this fact is funny, and it comes from a group of facts (not necessarily all funny), then the answer would be "is". Is there only one answer if it is considered grammatically, or could either one work?

In sentence #3, "are" seems to be the solution because "dice" is plural. However, I'm considering it as if the "group" is a solution; would I then be able to say "is" as if this one thing is my solution, or would it always have to be "are" because "dice" is plural?

Thanks in advance!