I hypothesise that internal consistency and agreement with our deepest moral intuitions are the two most important features of any ethical system. I'd like to hear suggestions of any other necessary and sufficient characteristics of a good ethical system. Does anyone have suggestions of books to read or thoughts to consider?

This is a pretty standard view in philosophical ethics, reflective equillibrium.

For a somewhat opposed approach, you might examine moral particularism (opposed to moral generalism), which roughly holds that we should make moral judgments about particular cases without (necessarily) applying moral principles, so while the particularist might care about coherence in some sense (when responding to moral reasons) they needn't be concerned with ensuring coherence between moral principles and between principles and our judgments about cases. You might separately wonder about how much weight should be given to our judgements about particular cases vs our judgements about general principles on a spectrum from hyper-particularism to hyper-methodism.

In terms of other characteristics of a good ethical system, I think it's worth considering that coherence doesn't necessarily get you very far. It seems possible, in principle, to have coherent views which are very bad (of course, this is controversial, and may depend in part on empirical facts about human moral psychology, alongside conceptual truths about morality). One might think that one needs an appropriate correspondence between one's (initial) moral views and the moral facts. Separately, one might think that it is more important to cultivate appropriate kinds of moral dispositions than to have coherent views.

Related to the last point, there is a long tradition of viewing ethical theorising (and in particular attempts to reason about morality) sceptically, especially associated with Nietzsche, according to which moral reasoning is more often rationalisation for more dubious impulses, in which case, again, one might be less concerned with trying to make one's moral views coherent and more in applying some other kind of procedure (e.g. a Critical or Negative one).

I am looking forward to trying to understand other people's ethical systems. Why do people make the decisions they do? What makes people change their minds? What allows people to ignore conflicting claims in their own beliefs?

There is a lot of empirical moral psychology on these questions. I'm not sure specifically what you're interested in, otherwise I would be able to make more specific suggestions.

Are there ways people perceive EA which are skin deep but nonetheless turn people off. eg a friend said "I don't think earning to give is good advice" even though most EAs today agree with this.

I think more applied messaging work about the reception of EA and receptivity to different messages would also be valuable to explore this and would likely help reduce risks which EAs run when engaging in outreach or conducting activities which are going to be perceived a certain way by the world.

I'm not sure that internal consistency should be the highest priority. If it is, that implies constraints on the applicability of a moral theory (ie some questions will be undecidable). Which may be fine, just be aware of that tradeoff.

Impossibility theorems are pretty common in mathematics. Arrow's Impossibility Theorem will apply to many ethical frameworks. Where it doesn't, other impossibility theorems (Godel, Gibbard, Holmstrom) are likely to apply. If nothing else, reading up on this class of theorems may be interesting.