You don't want to be on the hook for an injury taking place due to a malfunction.

Here is an example from ours:APPLIANCES.Appliances included in the rental of the premises, as noted in the rental agreement, such as a refrigerator, range, dishwasher, clothes washer or clothes dryer, belong to Landlord.If Tenant wishes to use these appliances, Tenant agrees to use them properly in accordance with manufacturer instructions and assumes all responsibility for their care and cleaning.In the event there is a malfunction of an appliance or an appliance is in need of repair, Tenant agrees to notify Landlord in a timely manner and in writing.If Tenant wishes to use their own appliances, Tenant agrees to contact Landlord in advance to obtain prior written approval and instructions, as the removal and installation of appliances must be done with care to prevent damage to the appliance and/or building.We always place water alarms near washing machines, hot water heaters, and under the kitchen sink.

For #4 - You need to be upfront with tenants and let them know that you are new to the house and you would appreciate any reports of broken/malfunctioning equipment, so you can get it in working order for them. 

Typically we wire lights in parallel so that a malfunction on one light doesn't effect the rest of the lights on the circuit (ie. old Christmas lights that go out when one bulb is unscrewed).

Landlord set up and control of the property Wi-Fi can provide many advantages, from becoming a small but steady source of extra revenue, to enhancing security (remote control and recording of gates, entries, exterior building & common area cameras); to saving time (remote entry control & rekeying of door codes); to saving money on maintenance (through instant water leak notification & advance indications of appliance malfunction); to saving both tenant and Landlord money on energy usage through remote timing and/or turn on/off all or some lights, appliances and devices;  to ensuring compliance with lease agreements (real time notice of smoking on property, excess people on property), and so much more.Did you know that for $300 (plus installation), a landlord can install a device that will record and report in real time the energy consumption of pretty much any electrical device on the property (https://sense.com/ ) ( https://www.schneider-electric.us/en/home/smart-home/wiser/wiser-app/ ).

Since it sounds like you do have a clause that requires them to have renters insurance, they should be good to go -- this really is on the tenant to make sure that they follow up with their insurance company for their claim; now, I will say (to Theresa and John's point), if the dryer caught fire and it can be proven that it was a technical malfunction, then that's a whole different situation.

I am wondering if I -- as the tenant -- am required to replace the items of personal property if they break/malfunction, or if I could simply leave the broken items in the property at the end of the lease.

It is also something you will be responsible for repairing when they malfunction.

If something breaks or is malfunctioning, I just swap the item out. 

I was just pointing out that a dripping shower or malfunctioning fixtures are generally not caused by tenants.