This is now a year old thread but after reading through the comments on the question, this almost certainly sounds like the building has developed bad wiring that is causing arcing, and constantly tripping the Arc Fault Circuit Interrupters.
The personal appliances may not be directly involved in this, other than completing the circuit, and allowing defective but idle building wiring to expose itself.
AFCIs are supposed to trip over and over until the real problem is fixed, because arc faults due to defective wiring will not go away on their own, and replacing the AFCIs themselves will not fix anything, unless it is in fact the AFCI that is arcing internally.
The building wiring can go bad all by itself even if it was originally installed and inspected properly, if there are external conditions that cause corrosion of the metal inside junction boxes, such as flooding, high building humidity, or exposure to salty ocean spray.
External forces such as earthquakes can also cause loose wiring arc faults due to pulling on wires and loosening of screw lugs and wire nuts, when buildings flex but don't collapse.
Really the solution is for the electrician (or homeowner if the jurisdiction allows it) to check the entire length of each faulting circuit, open the junction boxes from the breaker panel to all the endpoints, and check everything for loose wire nuts or loose screw terminals.
Since wire nut connections cannot be examined installed, they will all need to be removed to examine the wire ends and then retightened. All metal to metal wire contact and the inside of the wire nut should be shiny. Though if you're going through all this, it may make more sense to just replace all wire nuts with new and not bother reusing the old ones.
For push-in spring terminals, the wire end should be released, examined for corrosion or burn marks, reinserted into the spring terminal, and checked for firm anchoring with light tugging.
Outlet sockets and lamp fixtures can also arc inside where the plug or lamp base is inserted.
Check any device plugs for dark burn marks on the blades, or darkened scorch marks on socket faces where the blades go in. Replace both the burned wall sockets and the device plugs, not just one of them or the problem will restart.
Likewise check all lamp sockets for blackened or pitted / welded contacts and replace both the socket and lamp if found, because arcing causes unwanted heating and building fires.
If the problem persists after checking all easily examined junctions and terminals, the arcing can hidden be inside enclosed circuit devices that are not designed to be opened and examined, such as switches and receptacle sockets that are riveted/glued shut, or their hidden internal push-in spring terminals. It may be necessary to replace all of these circuit devices to see if the ACFI tripping finally stops.
Ideally if the circuit tripping occurs when some specific thing happens, this will likely lead you to the quickest resolution, such as "when I turn this room light on/off, the AFCI trips"... so in order of complexity, check the lamp socket, the lamp base, try replacing the sealed light switch with a new one, and then move on to examining all the building wiring end-to-end from the AFCI to the lamp, including in the lamp fixture itself.
Finally, if possible try to locate the cause of the external stress / damage factors, and attempt to prevent or alleviate them.
In some cases the building wiring may need to be changed, such as switching to using watertight and gasketed conduit to protect wiring that corroded due to high humidity exposure.
Wiring strain damage caused by earthquakes and building movement can be reduced by leaving coiled slack in the walls or junction boxes, so that the wiring has room to flex and move without pulling junction or terminal connectors apart.
Wiring and devices that are exposed to frequent vibration, position adjustment, or heating and cooling cycles can also develop loose and arcing connections, and may require design changes or more frequent inspection for damage.
Best Answer
As a renter, your "work" can only consist of trying to track down what the actual relationship of circuits to locations is (see if there's another spot that "looks like" it could have had a washer/dryer, for instance) and trying the devices you want to run on the outlets you want to run them on to see if a breaker trips, or not - given that modern refrigerators are usually small loads (mine does 50-60W unless it's defrosting at 200W) your main likely issue with overloading would be don't wash and dry and treadmill at the same time.
We have some previous questions and answers about tracking down circuits. Labelling is often outdated or wrong - just be systematic and make a note of what breaker affects each device location.
Any electrical work on a rental has to be done by a licensed electrician (as others already commented.)