In General Relativity Theory, mass can warp spacetime. However, in my view interaction only occurs between pieces of matter. Spacetime is not matter; how can it be affected by matter?
Answer
This is dangerously close to philosophy rather than physics, but your question has no answer because you appear to have a different concept of spacetime to the rest of us.
The concept of spacetime arose from special relativity and was proposed by Minkowski as a way to understand special relativity. There have been various questions on this site that expand on this. See for example What's the difference between space and time? for a discussion of why spacetime is a useful concept in SR.
The point of this is that we didn't start with an idea of spacetime and then try and work back to special relativity. Spacetime is a concept that emerged from SR. Exactly the same is true in general relativity, except that the metric is now variable rather than fixed and as you say in your question, the metric is determined by the presence of matter or more precisely the stress-energy tensor.
So in general relativity spacetime is defined as that which obeys the Einstein equation. That's why it doesn't make sense to ask how matter can warp spacetime, because that is the way that spacetime is defined.
You wouldn't be the first to find this slightly unsatisfactory and indeed Einstein himself was uneasy with an equation that had geometry on one side and matter on the other. Hence his famous comment about marble and wood. Nevertheless, as far as we know GR works perfectly. You are certainly at liberty to start with a different concept of spacetime and see where it leads, but you will find it hard to improve on GR!
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