When do you need to make a past participle agree with a noun?

Firstly, when the past participle is used as qualifying adjective, you must make it agree it with the noun. For instance, in “les choses faites”, you turn the past participle “fait” into the feminine plural. Secondly, when the past participle is used with the auxiliary “être” or with a stative verb (“être, paraître, sembler, rester…”) you must make it agree with the subject in the sentence. For instance “les équipes semblent concentrées” or “les filles sont parties”. Thirdly, when the past participate is used with the auxiliary « avoir », you never make it agree with the noun, such as in “les enfants ont couru très longtemps”. However, when the past participle is used with the auxiliary “avoir” AND there is a COD, which is the direct object, placed before the verb, you make the past participle agree the with the subject in the sentence. For instance, in “la pomme que j’ai mangée”, you must make “mangé” agree with the subject whereas in “j’ai mangé la pomme”, you do not.