I always unplug the tender when starting the Bike. I don't know if it would hurt to leave it on or not. But I do know that even in sub zero temps like we get here in WI my bike has no problem starting.
It won't hurt anything to start it with the battery tender connected, but why would you?
The battery tender doesn't put out enough oomph to make much of a difference on a cold start.
I think the better question is why is your new battery not firing up without being attached to the tender? I have used YUASA for years, and the only time I have a similar problem is when the battery is 3-4 years old. I ride all year long.
You can, but it might be a little hard on the charger. They aren't designed to pull that kind of amperage. Also, never leave it connected when unplugged from the wall. It will act as a resister and drain your battery over time..