If your RTC is failing the Y2K test and you use applications that require a Y2K compliant RTC, then it must be replaced to ensure no Y2K problems in the future. On most motherboards the RTC is not replaceable, so the motherboard must be replaced. The date changed correctly in question number 3. See question number 7. I have verified that the BIOS and RTC have no Y2K problems.
Problems can also come from the OS, applications, data, and date transfers.
In Windows you should also make sure that the long and short date formats use four digits for the year. It is up to the application vendor to ensure there are noY2K problems. To be absolutely sure, you should contact the vendor to see if your application is Y2K compliant. You may need to upgrade to more recent versions of some applications. Some utilities are available to test your applications and data for Y2K compliance.