Hm
No it's not usual and the continuing ESA payments will be treated as an over payment although not a recoverable one.
I would chase them again about the situation.
Providing you have a Fit Note starting from the day after your Tribunal you can back date your claim. Be aware that you can only make a new claim on the basis of a new condition or a deterioration in your existing conditions, such that an award is reasonably likely. You may be asked to provide medical evidence to support this.
Gordon
Nothing on this board constitutes legal advice - always consult a professional about specific problems