How to give anonymous users access to only certain nodes?
You can use the Rules module for this.

Here is a prototype of a rule (in Rules export format) to get you going:

{ "rules_limit_access_to_some_urls" : {
    "LABEL" : "Limit access to some URLs",
    "PLUGIN" : "reaction rule",
    "OWNER" : "rules",
    "REQUIRES" : [ "rules" ],
    "ON" : { "init" : [] },
    "IF" : [
      { "user_has_role" : {
          "account" : [ "site:current-user" ],
          "roles" : { "value" : { "1" : "1" } }
        }
      },
      { "OR" : [
          { "data_is" : {
              "data" : [ "site:current-page:path" ],
              "op" : "IN",
              "value" : { "value" : [
                  "some\/path1",
                  "some\/path2"
                ]
              }
            }
          },
          { "text_matches" : {
              "text" : [ "site:current-page:url" ],
              "match" : "node\/\\d+\/something",
              "operation" : "regex"
            }
          }
        ]
      }
    ],
    "DO" : [
      { "redirect" : { "url" : "no_access" } },
      { "drupal_message" : {
          "message" : "Access to the content located at \u0022\u003Cstrong\u003E[site:current-page:path]\u003C\/strong\u003E\u0022 is not allowed for Anonymous users.",
          "type" : "warning"
        }
      }
    ]
  }
}


Adapt the paths in it to fit your own needs, to reflect all the paths that anonymous users should not be able access. And for any access module you have enabled, allow access for anonymous users also (any paths that fit in the Rules Conditions will be blocked by the above rule, while any path that does not fit can be accessed by anonymous users). Obviously, using the "negate" facilities or Rules, you could also rework this prototype rule so that you only specify the paths allowed for anonymous users.