Does covariance and phenotypic variance have to be for the same trait to calculate narrow-sense heritability?
Yes. See below for more information.
Can I still calculate the narrow-sense heritability of AA [..] even though they're different traits?
No, you can't. A measure of heritability makes sense only for
- a given trait
- a given population
- at a given time
- in a given environment
[I'm also told that if any resemblance between parents and offspring is completely due to genetics, ie. without environmental effects, then $h^2$ is 0.5, but I'm a little confused with it.]
The above is not a exact quote from you but the original sentence made no sense. I rephrase what I suppose you tried to say.
The formulation is inaccurate and could therefore be misleading. It is wrong to say that any covariance between parent and offspring is only due to genetics. In reality, environments between parent and offspring often covary. There might also have covariance due to epigenetic reasons and potentially many other reasons.
You cannot make any claim about what value should $h^2$ takes for the special case where covariance between parent and offspring is solely due to genetic variance (or any other special case considering only genetic variance). You still need to know the relative importance of the genetic variance to the total phenotypic variance. If any of this is unclear, then you should keep on with the recommended reading below.
I recommend that you have a look at Why is a heritability coefficient not an index of “how genetic” something is? for an overview of the concept of heritability and common misconceptions.