Frustration With Fonts

If by "every time" you mean in any new document that you begin from scratch, yes. Modify a blank pages template and then select File > Save as Template

As for the paragraph styles. Those should retain within your document. It's possible that if you began with a blank template, or other template, there is some "hidden code" on the document forcing a different paragraph style. It's also possible that you are forgetting to update a selected style or you make updates that negate your original changes.

I use Pages a lot, and tried to recreate your issue, but was unsuccessful.

Me too. But, I created a template that incorporates my selection of typeface to match my corporate brand. That way I just choose my template (which is blank but the paragraph, headers, etc. are already defined)

Either open the template and save as. Or, create a new document and choose your template in the template selector