If I remember correctly, to apply for new style ESA you have to have paid enough tax for two or three tax years and have a disability or health condition which affects how much work you can do.
I claim the old ESA contribution based in the support group. Contrary to what most people think, CBESA does not have a savings limit. You can claim CBESA and IBESA at the same time, this then entitles you to free prescriptions and so forth, but then imposes an income and savings limit.
The warm home discount is an odd one, the qualifying criteria are set by the supplier. While with EDF I qualified even on CBESA because I got PIP, I am now with EON and don’t qualify.