Are PlayStation trophies viewable publicly on the web?
Is there a way to view my trophy progress publicly on the internet without signing in to my account?
Yes.
For example, https://psnprofiles.com (3rd-party site, it's not related to Sony) provides such data. Just search using user's PSN id. If user was never viewed on the site, it will take some time to fetch data, but all subsequent attempts to view this user will show data immediately.