The origin of this whole line of reasoning in Tibetan Buddhism comes from Dharmakirti's Pramanavarttika. There have been extensive commentaries and teachings on this from numerous great masters of Tibetan Buddhism. You can find commentaries on it from all the extant schools of Tibetan Buddhism. For an extensive set of discussions on this line of reasoning see this accomplished Tibetan master here.
Let's look at some of the verses in the Pramanavarttika where Dharmakirti sets out his definitions and arguments:
Without a transformation of the substantial cause
The substantial result would be unable to change.
[This is] like, [for] example, without the transformation 
Of the clay [there would be no transformation of a clay] vase, and so forth.
Without the transformation of some functioning entity [of the cause]
It is not reasonable that [the cause which] transforms some
[Of the result] is the substantial cause of that [result]
It is like an ox and a wild ox. 
The mind and the body are also like that.
Verses 61 and 62 of the Pramanavarttika
How should we understand this? I'm no expert so the following comes with a big warning that I might be entirely wrong. This is how I make sense of it...
Things necessarily arise from the transformation of a substantial cause into a substantial result. The example given is of unformed clay (the substantial cause) transforming into a clay vase (the substantial result). The nature of unformed clay is matter and the nature of a clay vase is also matter. That is... the transformation from unformed clay into a clay vase is the transformation of matter from one thing into another.
The second verse is a bit harder to read, but it is basically saying that if we notice that the result has undergone some transformation - say the clay vase has broken apart - then it is necessary to say that some functioning entity of the substantial cause also transformed - ie., the matter of the clay vase that was part of the functioning entity of the unformed clay.
Both of these are intuitive and in accord with how I perceive the world to work. With the arising of things there is generally a transformation that occurs. Unformed steel and glass and rubber transform into a car. Cars rust and break apart and transform back into unformed steel and glass and rubber. Glass breaks apart and transforms back into silica, et cetera, et cetera.
What's important for showing that the body cannot act as the substantial cause for the mind is that the latter can transform without a corresponding transformation of the body. Similarly, the mind cannot be the substantial cause of the body since the latter can transform without a corresponding change of the mind. Sometimes changes in the two coincide, but it is not necessarily so.
I'll leave the discussion about a higher being to others, but note that it is said in Tibetan Buddhism (and in many other traditions) that God's and so forth do eventually die and transform into beings of the lower realms.
Finally, I'll mention that I do have some qualms with this whole line of reasoning as it seems to be bound up with notions of the hypostatic existence of mind and matter. That is, if the reasoning is valid it can only be a conventional truth with the nature of illusion.
I hope this is helpful.