1. A director resgins by sending out a resignation letter to the company's Board. It's the company's responsibility to file form TM01
TM01 has to be signed by an officer of the company on the date it was signed, from what I remember. So I can file my own TM01 with Companies House for peace of mind, to know it's done and dusted, yes?
Do I have to formally notify the remaining directors?
Technically NO, because once resigned you're no more an officer of the company. Your resignation is effective the date you've resigned.