Reduced relative clauses help?

No, "lived" does not contain a passive meaning (a passive would be constructed with a form of "to be", and the subject would be the patient-- that is, the person or thing receiving the action), but it IS past tense. Since you note the difference between the past and present forms, I assume this is what you actually meant.
The use of the past tense in the prose suggests that the narration is recounting a situation from the past (the story itself could be entirely situated in the past, therefore all the verb tenses used would be a form of past tense). Presumably, it is unknown whether or not the children still live with the father, or it is known that they do not live with the father presently.

you need to envision your stepped forward grammar e book. We use participle adjectives immediately after nouns while they perceive or define the noun. that's corresponding to 'defining relative clauses' (ie relative clauses which particularly of giving extra tips define the subject of the relative pronoun.) for this reason those participle adjectives are many times called decreased relative clauses. eg: I counted the type of individuals waiting. (= who have been waiting.) We had to pay for the rooms used. (= that have been used.) some can be used earlier the noun. for this reason: international warming happens while gases noted as... yet: marketplace and transportation ought to offer gases that make contributions... (No participle, for this reason can't be 'decreased'.)

You could use either: lived with him, or are living with him. The point is that the kids are no longer with their mother.