I neede this work done so a new furnace could be installed, as the old one malfunctioned earlier this week, leaving me with an occupied property that had no heat.

Ask the tenants if there's anything that's broken, malfunctioning, or needs maintenance attention.

I personally would leave the current mower with the new tenants (and to have verbiage in the lease that holds you harmless for any equipment malfunction etc.). 

If water came up through the drains, complain to the City of Chicago about poor/malfunctioning  sewers.If water came in due to collapsed/clogged drain pipes.

Any damage caused by improper maintenance of the appliance or its malfunction will be repaired by the Landlord at the undersigned Tenant(s) expense.Tenant’s clothes washer and dryer will only be used by the legal occupants of the rental unit (as determined by the signed Lease Agreement, referenced above).

The board is located in the attic and about this time last year the tenant admitted to messing with the board when the A/C seemed to malfunction.  

Do you want tenants to let you know when something malfunctions or repair it themselves or just let it go?

Liability in this game is huge and if you work on a house yourself or you employ people to do it and it malfunctions then it's on you to fix it, whereas when you hire subs it falls on them to warranty their work.

In fact my out of pocket expenses were only to cover what COULD have been appliance malfunctions in two separate appliances and a shattered toilet.

I put about 5,000 rounds through my Ruger with only 1 malfunction and that was on a reload I made myself.