My lawyer is dragging my case, can I sue her for my retainer?

What is being done to drag it out? You don't sue for retainer. If there is unused funds then you are entitled to a refund. If its not given back then you can sue for the monies lost.

My case was settled for mediation and parenting coordinator, my 2nd lawyer ( the current one) cancelled the mediation, refused to file a contempt towards my soon to be ex for not complying with the judge ording us to attend parenting class ( I attended and on file in the court), refused to file for contempt towards my ex for missing not one but several dates to file his financials, refused to file for a child support or spousal support based on both of us live in the same house an he pays the mortgage yet he does not pay for food nor clothing nor gas, she took the side of my ex's attorney to cancel the parenting coordinator and elevate it to custody evaluation and send us to this incompetent unprofessional psychologist who has the reputation of being bias to the payer side which happened to be the husband side and when I told her in a private meeting in her office about his reputatuion, then in a conference with the judge she told the judge that I contest this psychologist because he is a man! So now, after all this what do you think I shall do with her?

I forgot to say that in the contract the retainer is non-refundable.

It sounds like they are doing work for you. The problem is that there is a process for everything. If someone misses a date to file something that isn't really contempt. There needs to be an order to compell before it can rise to contempt. If you don't like how you case is being handled then you have the right to get another lawyer. But it sounds like a lot is going on. How much was the retainer ?

$5,000 retainer and yes there were order to compele and never follow up on them.