Can a oregon officer issue a failure to stop at a stop sign and a careless driving ticket 4 months after a accident?

"Booked in jail" tells me this is a criminal case, and the traffic infractions you describe are problems -- but the least of the driver's problems. The driver needs a good criminal defense attorney, yesterday. On your specific question, yes, the police can charge an infraction like those you describe any time within 6 months of the incident date. The defendant should have gotten a criminal attorney months ago -- the less continued delay in hiring one, the better the likely outcome. I'm sorry you're going through this! You'll get more answers on the hit-and-run crime if you switch the question category to criminal defense. I almost did that for you, but your specific question is about the lesser infractions, so I'll leave it for you to decide if you'd like to put this in a different category.

The law says the officers shall not arrest persons for traffic violation, but may stop and detain them for the purposes of investigation reasonably related to the traffic violation, identification and issuance of citation. So unless there was some other reason for taking her into custody, the officer may have exceeded his/her lawful authority and violated her statutory and constitutional rights. Consult with an attorney ASAP.