(1) I could have broken his grip
Here "have broken ..." is in the present perfect tense, which denotes a present state due to a past action. By itself it would denote a past breaking of "his grip" resulting in a present state of being free from "his grip". Here it is modified by the modal "could", which allows the whole perspective to be shifted to a past point in time where there was a possibility of achieving that state (of having broken his grip).
(1) but then he could order me dragged back by the heels, and quite a few people would enjoy doing the dragging
Here "order" is by itself in the present tense, denoting an action at the present moment. It is modified by the modal "could", which as mentioned above allows a shift of perspective that to a point in the past where there was a possibility of the ordering. The author cannot use "could have ordered", because the perfect participle "have ordered" would denote a state, which does not semantically fit the context, since it is subsequent to the possible "breaking of his grip".
Perhaps it might be clearer to see how it would be phrased if it were a present thought running through your head now:
(2) I can/could break his grip, but then he could order me ... and quite a few people would enjoy ...
In the present tense we can use "can" for the first modal, but not for the second, because the ordering is semantically subsequent to the breaking of grip, and so necessarily set in the future. "Could" does not restrict the time of the modal statement, so it can be used in both cases, and is necessary in the case of past and future possibilities. Again we should not use "he could have ordered ..." because "have ordered" would denote a state, in this case a future state, which is not normally how we think of such scenarios.
Normally we consider it as a string of events. Here the first event is a breaking of grip, and the second is an ordering. In such strings of events it is natural to think of the second one in view of the first having happened, hence we can use the present perfect for the first event if it was in the past, but in the second situation that I showed above with the first event a present possibility, we also don't use the perfect tense because we are thinking about the possibility of the action, not the possibility of the resulting state.
Anyway modals in English are a minefield, and it's difficult even for native speakers to explain why certain constructions are correct and why others are not, and the nuances vary quite a lot over time and geographical region.
Your newly included examples are different because they are regarding what actually are true as of now (present state), such as "she has taken ballet lessons ...", where each of them is implying that if a certain condition had been fulfilled, then this true fact would never have been the case, and to convey this we use the perfect infinitive modified by the modal "could/would not".
The power of the Deity could have given ... , but then no true could have stood on the land ...
In this case you cannot use "no tree could stand", because "could" with the negative (such as "no tree could" or "could not") somehow means "was unable to", whereas "could" in the positive somehow means "would be able to". Don't ask me why English is like that! So "could" would be wrong here because the consequent of the conditional is dependent on the hypothetical condition of "could have given", and must be followed by a modal to denote a hypothetical result, but "no tree could stand" would mean a factual "no tree was able to stand", which is incorrect.
She could have broken her neck ... Then she would never have taken ...
As in the other case, the perfect infinitive denotes that it is actually true that "she has taken ballet ...". In this case, however, it is possible to say:
She could have broken her neck ... but then she would never take ...
"would never take ballet" by itself it would not convey that it she actually did take ballet. For example:
If we do not act quickly, we would never be able to escape.
This does not imply that we would actually "be able to escape" if we do act quickly. However, in your example the hypothetical situation of breaking her neck is in the past and so are the subsequent consequences, so the whole text does already imply that "she did take ballet", thanks to not "having broken her neck".