The close date comes and passes without notice despite me reaching out. 

I recommend that you look at Notice of Defaults, Vacants, Tax Liens or Probate. 

Nothing in this Section prohibits either party from filing a lawsuit and recording a lis pendens affecting the Property, before or after the date of written notice requesting mediation.

. - Who ever told you that only 30 days notice is required does not know CA law.  

It sounds like you have an un-permitted unit, which is essentially always a risk since there has been no notice to the AHJ that it exists and they can therefore initiate a code case to have it removed in most cases.

However, I noticed that as a Costco member, they also offer HVAC services, including the same product line (Lennox), along with a 10% Costco Shop Card if you go through them.Has anyone here used Costco for their HVAC needs?

Unless the medical condition meets the state criteria for early termination and that the medical notice does not appear to suggest that, you cannot terminate the lease outright/immediately but that you will begin to market the property for rental and upon securing an acceptable tenant, the lease can be terminated pending a satisfactory property walkthrough indicating no damages. 

This would likely take coordination with a tenant at some point - (ie - plenty of notice to them that they must vacate in order for you to sell the property)Since your current home is free and clear -  you could do a cash out refi or take out a new mortgage on the new house to be able to buy your paren't house and pay off the mortgage.  

At every point in history of the 15+ years of the site I've noticed the old timers complaining that the site is the worst it's ever been. 

Most mortgage companies will not call loan due if payments are current but you do run that risk and need to be prepared to payoff or refi at a moments notice.