Regarding one of the sinks in my home, the cold water line runs parallel with the heating line of my home and causes my cold water to be warm/hot for the first 30 to 60 seconds of use. Is it permissible to use it on Shabbos?

If you turn off the hot water system completely, so it’s not operational over Shabbos, you may use the cold water.

It is worthwhile to test it before Shabbos.