Legal Question in Civil Litigation in California

It's hard to answer your question without knowing what happened. Please re-submit your question with a bit more information. And please bear in mind that disqualifying a judge is much harder than most litigants realize.

"Is it necessary to file under CCP 170.3(b)(4) before filing under 170.3(c)(1)?"

This is not a yes or no answer. Read all of 170.3 because it answers your own question and you will know whether to file both, either, or none. If you're a lawyer and need help, then email me directly.

"would filing under either defeat a simultaneous CCP 473(d) motion?"

No bad motion practice will defeat a 473(d) motion. You do not have the proper ground to make a 473(d) motion which cites only a change in the law. For any lawyer reading this there is a 2010 case on this very point saying not to do that. To the laymen I will explain it this way. If 473(d) could be used every time the law changed then people would be re-litigating old cases all the time and the courts do not let people do that using 473(d) which was intended at obvious errors like failure to serve lawsuits on defendants but then entering judgments against them.

Do not hesitate to contact me further or submit a paid question as it is private advise.

Code of Civil Procedure section 170.3, subdivision (b)(4) is not a separate motion ground. Rather, it establishes a separate duty upon the judge. Code of Civil Procedure section 170.3, subdivision (c)(1) is the written statement required that a party uses if the judge has not recused himself under (b)(4).

None of those sections have anything to do with Code of Civil Procedure section 473, subdivision (d), however from a practical standpoint, a judge who you claim to be biased against you by reason of the procedures of section 170.3 is not going to be personally favorable to you on ANY motion.