Yes, a Privacy Policy is required even for apps that do not require sign-up. However, a sign-up screen is good as you can use the clickwrap method of getting active constent from users to accept your Privacy Policy.
The requirement of a Privacy Policy is due to sensitive permissions you app may ask.
You can find the list of those permissions here: https://play.google.com/about/privacy-security/personal-sensitive/
Even if you don’t need and ask those permissions, make sure you know what third parties you use in the app that might trigger this requirement (eg. AdMob).
You’ll need to add the URL to the Privacy Policy URL at Store Listing section. The link must be publicly accessible. Google will not host it for you.

A Terms of Service is not required, but it’s useful to add rules regarding user-generated content, unacceptable behaviour and so on. For example, it’s useful to add intellectual property clauses, ownership over your content and licenses over user-generated content clauses.