A: Generally no. You need legal immigration status. Tourists with an FMM are not normally issued a CURP. Exceptions exist for specific procedures (e.g., adopting a Mexican child).
Think of it as a Social Security Number (USA) or a National Insurance Number (UK), but much more comprehensive. While it was originally intended for citizens, it is now a requirement for any foreigner residing in Mexico temporarily or permanently.
Many people confuse these three identifiers:
Almost everyone interacting with the Mexican government or formal economy: