As Rolo says, that curving structure by your foot in the first photo is carbonised plant matter, which is locally common, often in surprisingly large chunks. It's arguably 'fusain', or fossilised charcoal, rather than lignite, or any of the other transitional states in the journey from wood to coal. As for the other photos, I'm afraid they're just typical of the iron-rich mudstones, siltstones, sandstones and concretions that make up the Ashdown Formation. Fairlight/Pett Level can be very frustrating at times - no end of rocks that look like they really should be fossils but aren't!
If you want to learn more about the area, there's an excellent little introduction by local expert Ken Brooks called 'Geology and Fossils of the Hastings Area'
It's available on line at https://www.ohps.org.uk/products-page/product-ohps/ohps-books/geology-and-fossils-of-the-hastings-area-by-ken-brooks/