From 711 until 1468, did the term Goth become generalized, and begin to refer to those descended from the inhabitants of Iberia before Arab and Berber arrival?

I don't think so.

Far from expert here, but my understanding is that the al-Quti name was only used by those who were descendants of Visigoths (several families used it, not necessarily related). This would be why Ismael Diadié Haidara, a descendant of Ali bin Ziyad from Timbuktu, released his biography of bin Ziyad under the title "The Last Visigoths".

Also, were there other terms to describe non-arab and non-berber people of al-Andalus in this period? Did these terms come freighted with religious connotations as well, or could they merely refer to ethnicity?

General terms for Iberian Muslims were, obviously, Andalusi (from Al-Andalus) and Muwalad ("mixed", transcribed as "Muladí" in Spanish). This were ethnic terms, not religious.

The Musta'rab ("arabized", transcribed "Mozárabe" or Mozarab), were Christians that lived under Muslim rule, and over generations had adopted part of the Arabic culture such as dress, language or dietary laws (eating pork was actually common in the first decades of Muslim rule, even by recent converts to Islam, but later fell, both among Muslims but also Christians, for example). Of these some continued to speak an Arabic-influenced Romance language, either as first or second language, which is called "Mozarabic" today, but was called "Latin" by its speakers. It disappeared around the 13th century.