How do I find out which Contrib module is compatible with PHP 7? Drupal.org misses a PHP version field. As far as I can see, I have to go through the issue queue or .info file for every third-party module to find out which PHP version is supported. Is that correct?
Sort of.

The PHP version isn't a required key for module .info files, and it's a minimum version of compatibility anyway. So a module could declare itself as requiring PHP5.6, but still work absolutely fine under PHP7. Or it could not declare a minimum version of PHP at all, and still work under PHP7. Or it could declare PHP5.6 as minimum, but not work at all in PHP7. You get the idea.

So yes, you need to check on a case-by-case basis to find out which will work under a specific version of PHP.