You are probably using the dot . as multiplication, but in WL it means scalar product of vectors. For example {1, 1} . (0) does not give zero. Plain multiplication is made with *

This is a very common stumbling block for new users. Any function that ends in Form is for presentation purposes. What they do is wrap your expression in a way that tells the front end how to display the expression. It's best to avoid doing computations directly on the presentation form. So, when you did this

minkMetric =
 {{1, 0, 0, 0},
   {0, -1, 0, 0},
   {0, 0, -1, 0},
   {0, 0, 0, 1}} // MatrixForm

you didn't actually set minkMetric to a 2-dimensional list, but to an expression with head MatrixForm that itself contains a 2-dimensional list. That MatrixForm head tells the front end to "paint a pretty form" of the matrix. You can see this explicitly using FullForm:

minkMetric // FullForm
(* MatrixForm[List[List[1, 0, 0, 0], List[0, -1, 0, 0], List[0, 0, -1, 0], List[0, 0, 0, 1]]] *)

The idiom I use looks like this:

minkMetric =
  {{1, 0, 0, 0},
   {0, -1, 0, 0},
   {0, 0, -1, 0},
   {0, 0, 0, 1}};
minkMetric // MatrixForm

Now we've assigned the 2-dimensional structure to the symbol minkMetric and then just displayed it in a pretty form. You can see that minkMetric now is just a plain matrix.

minkMetric // FullForm
(* List[List[1, 0, 0, 0], List[0, -1, 0, 0], List[0, 0, -1, 0], List[0, 0, 0, 1]] *)

Now you can access the items:

minkMetric[[2, 3]]
(* 0 *)

As for your follow up question, I would need more information.

You'll get better answers if you post code rather than pictures, because that allows people to copy-paste your code into their own notebooks to play around with. All I can do is make a guess based on your picture. My guess is that those are Dot products, and Dot doesn't evaluate when the arguments are not of compatible dimensions.