I think you will find that you can only appeal again on a point of law - in other words, if the Tribunal is deemed to have mis-interpreted the relevant law in some way. You need some specialist help with this - local CAB or other free welfare benefits advice service. Meanwhile, claim whatever you can as advised above. Also, if you don't live with your father & are not getting housing benefit (if you pay rent) or council tax benefit claim those from the local authority.
The appeal tribunal you went to would not really be concerned about your caring responsibilities - their role is limited to determining whether (in their judgement) you are fit to work. They might look at the caring work you do in that context but not as part of a justification for you remaining on ESA.