In PHP, how do I check if a function exists?

Using function_exists:

if(function_exists('my_function')){
    // my_function is defined
}

Cracks knuckles

Technically the syntax is "correct" (it won't generate a fatal error) but the semantics of PHP render it effectively meaningless in its current form. Let's look at a few things first, namely how PHP handles the assignment of named functions to variables:

php > echo shell_exec("php -v");
PHP 5.4.16 (cli) (built: Oct 30 2018 19:30:51)
Copyright (c) 1997-2013 The PHP Group
Zend Engine v2.4.0, Copyright (c) 1998-2013 Zend Technologies

php > function speak($arg) {echo "{$arg}n";}
php > function give($arg) {return $arg;}
php > $speak = speak(4);
4
php > $give = give(4);
php > var_dump($speak);
NULL
php > var_dump($give);
int(4)

The function itself is executed upon assignment and its return value (NULL or otherwise) is assigned to the variable. Since we're only assigning the return value of a function's execution, trying to use this variable as a function name has no use:

php > $speak(4);
php > $give(4);
php >

Let's contrast this to the assignment of an anonymous function (a.k.a. a 'Closure'):

php > $min = 1; $max = 6;
php > $checkName = function ($value) use ($min, $max) {
php {   echo "value: {$value}n";
php {   echo "min: {$min}n";
php {   echo "max: {$max}n";
php { };
php > var_dump($checkName);
object(Closure)#1 (2) {
  ["static"]=>
  array(2) {
    ["min"]=>
    int(1)
    ["max"]=>
    int(6)
  }
  ["parameter"]=>
  array(1) {
    ["$value"]=>
    string(10) "<required>"
  }
}

Unlike some other languages, a closure is represented in PHP by an actual Object. Variables inside the 'use' clause are imported at the time the Closure was created; function parameters (i.e. $value) have their values captured when the Closure is called (hence why we see it noted as a required parameter and not a static value). The semantics of references within Closures aren't worth considering right now but if you want further reading, goat's answer to this question is a great start.

The major takeaway here is that the Closure's assignment to $checkName did not execute the Closure itself. Instead, $checkName becomes a sort of "alias" we can use to reference this function by name:

php > $checkName("hello ");
value: hello 
min: 1
max: 6
php >

Given how loose PHP is about the number of function parameters passed, a zero-parameter execution returns expected results:

php > $checkName();
value:
min: 1
max: 6
php >

Now let's take it another level deeper and define a function within a function:

php > function myOuterFunc($arg) {
php {   function myInnerFunc($arg){
php {     echo "{$arg}n";
php {   }
php { }
php > $myVal = myOuterFunc("Hello ");
php > var_dump($myVal);
NULL
php >

By now this result should make sense. Functions do not execute unless explicitly called; just because we call myOuterFunc doesn't mean we execute any function code defined inside of it. That's not to say that we couldn't:

php > function myOuterFunc($arg) {
php {   function myInnerFunc($arg){
php {     echo "{$arg}n";
php {   }
php {   myInnerFunc($arg);
php { }
php > $myVal = myOuterFunc("Hello ");
Hello 
php > var_dump($myVal);
NULL
php >

Which brings us back around to what is essentially your question: what about a named function inside of a Closure? Given what we've now discovered about function execution, we can generate a series of very predictable examples:

$min = 1; $max = 6;
$checkName = function ($value) use ($min, $max) {
  function question(){echo "How are youn";}
  echo "value: {$value}n";
  echo "min: {$min}n";
  echo "max: {$max}n";
};
php > $checkName("Hello ");
value: Hello 
min: 1
max: 6
php >

As expected, the named function's code inside the Closure is not executed because we have not explicitly called it.

php > $min = 1; $max = 6;
php > $checkName = function ($value) use ($min, $max) {
php {   function question(){echo "How are youn";}
php {   echo "value: {$value}n";
php {   echo "min: {$min}n";
php {   echo "max: {$max}n";
php {   question();
php { };
php > $checkName("Hello ");
value: Hello 
min: 1
max: 6
How are you
php >

Explicitly calling the inner function works just fine, provided we define that function before we call it:

php > $min = 1; $max = 6;
php > $checkName = function ($value) use ($min, $max) {
php {   echo "value: {$value}n";
php {   echo "min: {$min}n";
php {   echo "max: {$max}n";
php {   question();
php {   function question(){echo "How are youn";}
php { };
php > $checkName("Hello ");
value: Hello 
min: 1
max: 6
php >

php > $min = 1; $max = 6;
php > $checkName = function ($value) use ($min, $max) {
php {   echo "value: {$value}n";
php {   echo "min: {$min}n";
php {   echo "max: {$max}n";
php {   function question(){echo "How are youn";}
php {   question();
php { };
php > $checkName("Hello ");
value: Hello 
min: 1
max: 6
How are you
php >

So to the point of your questions then:

1. Yes it's legal and what you're attempting is possible but semantically meaningless in its current form.

2. Any named functions inside the Closure definitely do not reside in the global namespace, they are within the scope of their defining Closure. FWIW, the term "members" typically refers to class variables (usually called "properties" in PHP). While Closures are an Object and let you duplicate the functionality of instance variables found within classes, they should not be confused or construed with classes in any way.

3/4) Not how you're trying to use it, no. Nothing outside of the Closure has any concept of the functions inside; if in calling your Closure code, said code performs operations using the inner functions, then they will see the light of day. But there is no immediate way to reference those inner functions as if they were defined outside of the Closure's scope.

In other words, the only way you'll get those inner functions to execute is if a) that code is specifically executed by the Closure's code and b) you execute said Closure code.

Hope this helps.

What about using Youtube's API?
After all, that would mean using some official, which is less likely to change than going with parsing some HTML page.

For more information: YouTube APIs and Tools - Developer's Guide: PHP

The Retrieving a specific video entry seems quite interesting: if you send a request to an URL like this one:

http://gdata.youtube.com/feeds/api/videos/videoID

(replacing "videoID" by the ID of the video, of course – "GeppLPQtihA" in your example), you'll get some ATOM feed if the video is valid; and "Invalid id" if it's not

And, I insist: this way, you rely on a documented API, and not on some kind of behavior that exists today, but is not guaranteed.