No, tripping the breaker won't harm your Tesla vehicle. The breaker is designed to protect your home's wiring from dangerous over current / overheating that can start a fire. When a breaker trips it is an indication that something caused the circuit to be overloaded. Breakers sometimes fail to trip when they should to protect the home and wiring. Always determine the cause of the overload condition, correct it before continuing to use the circuit.
The Tesla Mobile Connector kit comes with a 5-15 plug adapter that fits a standard outlet. The default charging amperage that the Mobile Connector will automatically set with the 5-15 adapter is 12 amps. This also happens to be the maximum allowed charging amperage for a 15 amp rated circuit. (I don't know why Tesla chose to default charging with the 5-15 power plug adapter to 12 amps. General Motors plug in and electric vehicles (Volt and Bolt) that come with a 120V/240V charging adapter default to charging at 8 amps with the 12 amp setting being a user set option.)
The issue, as you have discovered, is that a 15 amp circuit will trip when you attempt to charge at 12 amps and run an appliance such as a microwave oven or toaster that is connected to the same circuit. You should only use the 12 amp default charging rate with a 15 amp circuit if you are certain that the circuit is not being shared with other equipment such as a refrigerator, freezer, microwave, toaster etc.
You can lower the charging amperage from the Tesla charging screen or in the Tesla app. An 8 amp setting would enable charging at ~3 miles per hour instead of ~4 miles per hour and would be less likely to overload the circuit and trip the breaker.
While you are waiting for your dedicated charging circuit to be installed you can use the Plugshare phone app and web site to locate public charging stations near where you live, work and shop. You should have received a Tesla SAE J1772 charging adapter with your Model Y. The J1772 adapter enables you to charge yoru Tesla Model Y at any public Level 1 or Level 2 charging station that has a J1772 charging cable connector.
You can locate Tesla Supercharger locations and DC Fast Charge stations using the Tesla Navigation system. To charge using a DC Fast Charge station such as Electrify America, Chargepoint, Blink or EVgo you would need to purchase the Tesla CCS Combo 1 adapter from the Tesla.com online store. You can also purchase similar CCS1 adapters from Amazon.
