Anyway, we decided on a refundable deposit, proof of it being a "service" animal, and an addendum that she sign and return that basically protects both parties.

My mentor highly recommended against a 50/50 split since he has had issues in the past where both parties had opposite opinions which ended in a stalemate and no action being taken. 

I would like it the benefit both parties but most importantly the best way to benefit her. 

California is a lien theory state... so based upon that, a mortgage or deed of trust executed by one joint tenant would not sever the tenancy unless the property was sold at foreclosure prior to the death of the party who incurred the lien.If the other owners executed the HELOC they would be pledging their own interests as collateral... which I doubt is the intention.

Post notice of past due rent to get the eviction party started.  

One party can pay it or it can be split between the buyer and seller-spell it out on the contract and get more info from the closing attorney or title company in your area.

Anyone with advice on a way to structure that makes both parties comfortable? 

I definitely need to implement a way that makes it beneficial for both parties.

But that’s the part I am nervous about, because I am unsure of how it works and what my mortgage will be after.

Figure out which each part is worth to you both and then calculate who gets what % based on what each of you put in.