This looks like a job for $.map!

var data = {
    "category": [{
          "id": 28,
          "name": "Dogs"
      },
      {
          "id": 14,
          "name": "Cats"
      },
      {
          "id": 878,
          "name": "Sheep"
      }]
}

var cats = $.map(data.category, function(v){
    return v.name;
}).join(', ');

This works on the JSON document given in the question:

$ jq -r '.roots.bookmark_bar.children[]|.children[]|["\"\(.name)\"",.url]|@tsv' file.json
"The title is here"     https://the_url_is_here
"The title here"        https://url_here

This accesses the .children[] array of each .roots.bookmark_bar.children[] array entry and creates a string that is formatted according to what you showed in the question (with a tab character in-between the two pieces of data).

If the double quotes are not necessary, you could change the cumbersome ["\"\(.name)\"",.url] to just [.name,.url].

To trim the https:// off from the URLs, use

.url|ltrimstr("https://")

instead of just .url.

If i understood correctly, you can do the following:

var json= {"Title":"Movie1","Year":"2013"};
var result="";
for( key in json){
  result+= json[key];
}

You don't have to know the number of properties or it's names before hand. This should work for simple scenarios.

It sounds like your @RemoteAction should return multiple addresses where each address looks like "city, postcode, citycode". For that to work I would expect that the JSON returned would be an array of the objects you post in the question not a single object.

The easiest way to parse such a structure is to submit an example of it to JSON2Apex and use the code that is generated.

The return type of your @RemoteAction method remains List<String> but each item in the list is:

calOut.add(p.city + ', ' + p.postcode + ', ' + p.citycode);

assuming p is a reference to the properties field of one of the returned and parsed objects.

PS

The names here depend on what the JSON fields are called but the code would look something like this:

List<String> calOut = new List<String>();
JSON2Apex parsed = JSON2Apex.parse(res.getBody());
for (JSON2Apex.Feature f : parsed.feature) {
    JSON2Apex.Properties p = f.properties;
    calOut.add(p.city + ', ' + p.postcode + ', ' + p.citycode);
}
return calOut;

Note JSON stands for JavaScript Object Notation. JSON is always a string. It is a string representation of your encoded data. There is no such thing as "a JSON Object".

Don't attempt to write JSON by joining strings and other values together. Instead build an object and pass it to JSON.stringify -

const dept = "sci"
const avg = 77.09

// make an object
const myobject = { courses_dept: dept, courses_average: avg }

// use JSON.stringify to encode myobject to JSON
const myjson = JSON.stringify(myobject)
console.log("encoded", myjson)

// test JSON.parse to decode the JSON and get the object back
console.log("decoded", JSON.parse(myjson))

encoded {"courses_dept":"sci","courses_average":77.09}

decoded {
  "courses_dept": "sci",
  "courses_average": 77.09
}