I cant say the 2 second power up at coil will hurt anything. It is not normal as I understand it. Could be something different in the PCM than what I am used to seeing. I cant honestly answer that.
The ASD relay powers the coil and fuel pump /injectors etc, the ECM controls it, I dont see it hurting anything.
If it is running I would not worry about it.
Glad it is burnin gas again.