It's not your job to baby the tenant.

Was anyone rushed to the hospital recently to have a baby...?  

One had a baby while in the unit and she just bathed the baby in the sink. 

And again those are situations that you as an individual work out in your own way given your financial condition and how you feel about a certain tenant.. my classic was the tenant who told my PM they just had a new baby ( which I believe they did) record cold winter they could not afford heat and rent could I do something.. so of course I say sure.. don't pay two months and add it to the rent and pay over the course of the coming year.. end of two months they up and left  LOL. and for sure I can see how investors that are highly leveraged with little to no real reserves might react differently than those who might own the real estate free and clear or if they missed a month or two of rent its just no biggee 

They were old-school depression babies and did most of their own maintenance even though they were in their 70's and 80's.They were soft-hearted with tenants and the tenants learned to go around me when they thought I'd say no (like little kids) and waylay the brothers in the parking lot when they'd show up on a Saturday morning, to get the "owner" to agree to something instead of asking me first. 

Tenant now is threatening to "talk to his lawyer" for the pain he / his family (wife + baby) have been through, and that he is "collecting the evidence". 

This baloney with the property manager disagreeing with the plumber's rate and all of you having team shouting matches with the plumber's wife -- this is all baby-talk baloney.

I think the way around that is to just advertise that this is the rent for non-pet owners, and this is the rent for pet owners, and if I find out you have a pet, the rent agreement would say that both parties agree that it will be assumed that the pet was there from day one, and the higher rent would be owed back to the beginning of the lease.Either way, tenants will try to sneak in pets.The owner I worked for would come on weekends and haul off garbage to another of his properties that had a big dumpster - he was too cheap to pay for twice a week garbage pickup (depression baby). 

My second question is, "How can I provide better service to my tenant without babying them?"

@Clinton Fisher great comment and I’m fascinated by that baby boomer downsizing.