Is it possible to use an adjective when there is no noun? I have seen this done, such as "sekva" on lernu.net. What's the rules around this? My belief was you used an adverb ending when there was no noun.
You are correct that under most circumstances, you don't use an adjective when there is no noun. Adjectives modify nouns, so if you're not modifying a noun, you almost never use an adjective.
There are some circumstances when a noun is implied. In that case, you can use an adjective. A common example of this is "la angla" which implies "la angla lingvo."
Looking at the "sekva" button on Lernu.net, I would say that this usage is OK because the noun is strongly implied. It means sekva paĝo or sekva alineo.
In rare circumstances, and adjective and be used to modify a whole phrase if doing so would resolve an ambiguous reading. I'm still looking for the reference and will update this answer when I find it.
Adjectives modify nouns. Sometimes the noun is implied.
Mi parolas la hispanan (lingvon).
Mi legas la verdan (libron).
Alklaku ĉi tie por iri al la sekva (paĝo).
A noun phrase (article adjectives noun) can do without noun. This Esperanto copied from its national languages.
Or otherwise: it is an adjective when a noun is implied, certainly when there is an article. Or when one can add a noun.
For a language feeling: when the word relates to a subject or object it is an adjective.
the next (the next one) la sekva (tio sekva / la sekva ...) at three (at three o'clock) je la tria (horo)
And for a difference between adjective and adverb:
mi lernas la anglan mi parolas en la angla mi parolas angle (relates to 'parolas') mi angle parolas mi lernas pri la angla (kulturo) WRONG: mi lernas angle (angle does not relate to 'lernas')