If my outbound flight is cancelled and I don’t like the alternative offered so I buy a new one-way ticket on another carrier, will the original airline cancel my return flight?

In most cases, yes. If the original airline with the cancelled flight (Airline A) agrees to accommodate you on another airline (Airline B) because they are unable to do so on their own airline, then your return would be protected. However, if none of the alternate arrangements offered by Airline A will work for your needs and you simply buy a new ticket on airline B to get where you need to go, Airline A’s system will automatically cancel the return when you fail to show up for their re-scheduled outbound flight. In some cases you may be able to prevail upon Airline A to keep your return flight intact, but this takes some effort and runs the risk that the airline may convert the return flight to a one-way ticket, which is often a higher price than the original round-trip. A safer bet would be to cancel the entire ticket on Airline A and hold it for future use and buy a new roundtrip ticket on Airline B for this trip.