There is a real risk that you could lose your ESA if you go to appeal. I suggest that you get face-to-face advice to discuss this and help you decide how secure your current award is. The tribunal service can ask for a 'medical', I don't know how often this actually happens.
Don't forget that HRM DLA is is awarded because the claimant is 'unable or virtually unable to walk'. However, the descriptor for ESA is 'mobility' and you can be granted zero points here is the assessor decides that you would be able to self-propell a wheelchair (even if you don't have and would not benefit from one)!
As you say, the date on the letter seems to be random, this is something we see a lot and I can only guess that it has something to be with DWP internal systems.
Nothing on this board constitutes legal advice - always consult a professional about specific problems