In Washington State can mediation delay the final court date?

Know that just because there is a 90-day waiting period in Washington, it does not mean you are entitled to a divorce in 90 days. The only people whose divorces are finalized in 90 days are those who agree on the terms of their divorce. People who choose to contest the issues experience much longer periods between the petition and the final documents. There is no "entitlement" to three months. There is no "legal three month time frame." The 90 days is the shortest amount of time from petition to decree of dissolution, but in many, many cases, it can take a year, two years, even longer. Just know that the more you fight, the more it costs, the longer it takes.
Re your mediation, if you have a trial date set, you may want to notify the court that you have mediation and may not need the court date if you and the other party are able to resolve all your issues peacefully, or, in the alternative, you may need the trial date to be continued. If you require the latter, you will need to file a motion for continuance of the trial date and schedule it for hearing before the court. The court will not continue a trial date when someone comes in two weeks or less to spare before trial. The court has many cases it needs to set for trial and it needs enough time to fit another case in your trial time if you are not going to be in trial on that date.