I'm wondering whether it's possible to edit the "new" document settings so that my text styles are already present? Ideally, with a page of symbols and several pages already created and named too. Any ideas?
Could you not use a template? Create a Sketch file with your pages, styles, symbols etc. and then goto File > Save Template... When creating a new document goto File > New From Template and select your template.
File > Save Template...
File > New From Template
Indeed, that is basically what a Template is intended to do for you