They told me not to do any work yet while this is going on and in case of blockage, just call the plumber to open the clog. 

i had the sewer cleaners (plumbers) go to the house and for $125, they took the 2.5" cap at the bottom of the stack by the kitchen/basement bath.they said i have a "sink line" blockage and cleaned it. i had to put a new cap back since they didn't have one on them. while working on it i wondered how soon until the cast iron piping fails in each of the houses that was built in 1960 and how much that tear out will cost.... !!

They actually CAUSE blockage as the tenant gets careless and tosses everything down the sink :sigh:

Some blockages are the landlord’s responsibility (poor maintenance, roots, broken line, etc), but if we find the blockage was due to paper towels, rags, diapers, or anything else that should not be flushed down the drains then the landlord might be able to hold the tenant responsible for the cost of the cleaning.

Of course that will be, what matters is how much blockage the main sewage pipe has going to the sewer line out back, that's the only thing that matters. 

Is there any kind of collapse where there are jagged edges that will catch things and cause blockages?

Such as a blockage that should be the tenants issue but after repeated snakes and hydro jetters you need to dig under slab and clear blockage.

We had video verifying that the blockage was in the city easement, but the city would not even send someone out until we had excavated.  

Air conditioning in the summer, heat during the winter, broken windows or doors, refrigerators, drain blockage etc...are all emergency items where I am and should be treated as such.If HVAC companies are booked three weeks out, then you need to find one that can come ASAP. 

If the Tenant was living there prior to your ownership, they should be responsible for any blockage resulting from their actions or misuse.