Is a lawyer allowed in the mediation room with me and my ex?

Short answer: yes.  I presume you mean through the Court's "Family Mediation Center" or FMC.  For a little background, see http://willicklawgroup.com/mediation-services/.  If so, the FMC rules permit lawyer involvement, although if it makes you uncomfortable, they will normally accommodate you by placing the parties in separate rooms or using some similar protocol.  If it is private mediation not through FMC, both parties can insist on whatever ground rules (including "no lawyers present") that they wish and can agree to.  Really, your best tack may be to hire counsel.