Most of the questions you ask will be determined by the arbitration agreement. Review the arbitration agreement. It should state the arbitration administrator, the location where the arbitration would be held.
You have described a month-to-month tenancy under an oral lease. You can give a tenant 15 days notice before the end of the payment period that the lease is terminated. If they do not move out after the notice you may evict
HOAs can and will foreclose on your property. They can negotiate in that manner because they have the power to force the sale of your house. if you cannot work something out with the HOA, consult an attorney as soon as possible. Has a foreclosure action been filed? Has a lien been recorded?
Who did the house belong to and was there a will? If you are one of the heirs then you can open an estate and have your-self appointed as PR. I think an unlawful detainer action is probably the better choice instead of eviction but it depends on whether the ex-husband was paying rent.
The foreclosure law firm will keep moving ahead with the foreclosure unless yo take some action in the foreclosure lawsuit to stop them. You need a foreclosure defense attorney to help you immediately. The next step the foreclosure law firm will probably take is to set the motion for summary judgment for hearing approximately 30-60 days out. If you do not have an attorney, they are likely to get a sale date scheduled a month later and your house will be sold. Do not rely on the modification...
The landlord does not have an obligation to keep you in the property. If the landlord wants to put a family member in the property, there is nothing wrong with that. The landlord just has to follow the law when she terminates the lease and it appears she has done so. An attorney should review the lease to determine if you have any other rights.