No, "scratchaway" does not mean "quarrelsome". The cat names don't really have any meaning in everyday use. But we can attempt to deconstruct them using basic English patterns and see an association.
The writer is describing that cat as quarrelsome (~ disposed to quarreling, fighting). What do cats do when they fight? They scratch (~ attack with their claws). So, we get the idea that this cat tends to scratch.
Furthermore, away has a kind of deliberate, eager, or steady feeling to it, as in
6 : steadily onward : uninterruptedly
// clocks ticking away
6 incessantly or relentlessly; repeatedly:
He kept hammering away.
We combine these two and come up with Scratchaway.
And just to address the other name, the other cat is described as quickly flying into a temper (~ quickly and suddenly becoming angry). We associate having a (bad) temper, or broadly anger, with fire or heat
2 habit of mind, especially with respect to irritability or patience, outbursts of anger, or the like; disposition:
an even temper.
3 heat of mind or passion, shown in outbursts of anger, resentment, etc.
We can further associate this feeling with a pepper (~ spicy, hot, heat, fire, etc.).
A pot is a cooking container/utensil, but we could use pot to describe something being contained in a pot, or as if in a pot:
- literally: There is a pot of peppers on the stove (= There are peppers in a pot and the pot is on the stove).
- figuratively: He is a walking pot of peppers* (= He is very rageful).
We combine these two and get Pepperpot.
Note that these names are odd, uncommon names for pets. They were crafted for poetic effect. I wouldn't take them seriously.
Also, compared to your previous question regarding Sootikin (the third cat), using away and pot in this manner to construct names is uncommon. At least -i/y and -kin are commonly, or somewhat commonly, used to create pet names (though, -ikin is possible, but less common).
*Constructed for illustrative purposes. Not actually a common saying.