Le cas de non-accord ?

Actually, there is one and only one rule (with no exceptions and special cases) that you need to remember when it comes to matching the particple in passé composé. It is not the one that is usually taught (even here on kwiziq), which has several exceptions. If you're curious, you can just skip to the bottom of this post. But first, let's look at your question in detail.

Aurélie s'est disputée avec sa soeur. -- Aurelie had an argument with her sister.

The COD of this sentence is the reflexive se. And since it refers to Aurélie, you need the female version of the participle. Sa soeur is not the COD. One easy way to see this is to note that it is preceded by a preposition (avec). A COD isn't preceded by a preposition, it stands by itself.

Now let's look at one of the other examples you list.

Ils se sont lavé les mains. -- They washed their hands. Here the COD is les mains. The reflexive se is an indirect object. Since the COD comes after the participle, the participle does not match the COD in number and gender. To make it absolutely clear that se is the indirect object, we just need to rephrase the sentence as a question: 

À qui ont-ils lavé les mains? -- Whose hands did they wash? The phrase à qui (which stands for the original reflexive se) is clearly the indirect object as indicated by the preposition à.

Let's rephrase the sentence so that the COD moves in front of the participle, which forces the particple to match the COD in number and gender:

Ce sont les mains qu'ils se sont lavées. -- It's their hands that they washed.

So, actually, the rule that reflexive verbs always form the passé composé with être instead of avoir is not the best rule, since you also need to learn all the exceptions to this rule.  And you also have to remember that the participle accords in number and gender even when using avoir, in cases where the COD precedes avoir. 

The one and only rule that governs how the particple in passé composé is matched is this:

If the COD comes before the participle, the participle matches it in number and gender, regardless of whether the auxiliary verb is être or avoir.

Hi Andrea,

Perhaps you are confusing different aspects of past participle concordance?

Aurélie ________ avec sa soeur.  If the question was about Aurélie bickering or argueing with her sister then the verb se disputer would be appropriate. Sa soeur is indirect after the preposition "avec"

The passé composé conjugation should be elle s'est disputée where the past participle agrees with the subject Elle (Aurélie).

So the issue to get clear in your mind is whether the verb is required to be reciprocal or reflexive? 

The other point that may be troubling you is the situation with avoir usage when the direct object appears before the verb. In that construction, there needs to be concordance with the past participle.

Hope the above helps somewhat?

Bonne continuation

Jim