Do you intend to distribute your theme to others, or is it for your own specific site? For your own site, do whatever works for you ?? For others, you should consider usability. Ideally, users shouldn’t need to be concerned with styling. They would add content and it’d look awesome right away with no additional effort. Typically, the more people fiddle with content appearance, the worse it becomes. Pages with 8 different fonts in multiple colors and sizes will look amateurish.
All basic styling should be in style.css. If any content specific styling is needed, the styles available through the block editor UI should be relied on. Those are pretty extensive as they are. You shouldn’t try to manually enter your own style attributes via the code editor. As a last resort, one could add an additional CSS class and related rules in Additional CSS of the customizer. Then the same class/style can be easily reused elsewhere.
The specifics for flushing the browser cache depends on the browser. Do a general internet search to see instructions for your particular browser. On the server, if you have a WP caching plugin, how to flush it again depends on the plugin. There ought to be a settings page button or admin bar item to clear or flush the cache. If your host has setup caching on their accounts, there’ll be some way to flush or clear the cache from your hosting account. Similar for 3rd party caching like Cloudflare, there’d be a way through the related account.
Your browser cache is most likely the cause of stale data. Your host providing server caching is least likely.
