Q: Why does this array-like object behave like this? By setting the prototypal methods of an object to Array methods, the object behaves like a hybrid between an object and an array. Below is a simple example:


function Foo() {}
Foo.prototype.push = Array.prototype.push;
Foo.prototype.forEach = Array.prototype.forEach;

var foo = new Foo();

foo.push('abc');

foo.length; // = 1 as expected. But wait, why isn't foo.length undefined? How/when did this property get attached to foo? 

foo[1] = 'def';

foo.length; // still = 1. But foo={0:'abc',1:'def'}, Why not =2?

foo.forEach(function(item) {
  console.log(item)
}); //shows only'abc' and not 'def'

foo.push('ghi');

foo.length; // = 2, and now foo = {0:'abc', 1:'ghi'}. So it overwrote the key=1, which means its accessing the same location, but the first approach did not change the length ( didn't become a part of the array ) why ?

foo.forEach(function(item) {
  console.log(item)
}); //now shows 'abc' and 'ghi'


Why is all this weird behavior happening, and why isn't it good to mimic Arrays like this?

A: How is the length property set?
When you invoked the Array#push, or in this case, Foo#push method. Per the ECMAScript 2015 Specification:

22.1.3.17 Array.prototype.push ( ...items )
[...]


*[...]

     [...]
     d. Let len be len+1


*Let setStatus be Set(O, "length", len, true).


So when you invoke the function, it automatically sets the length property if it doesn't exist, to the value len which is incremented every time you push.
Why isn't the length 2?
Now, when you directly set an index in the array, the length property is not updated. That's because arrays in JavaScript are exotic objects which internally increment length when you set properties directly. Per the specification again:

9.4.2 Array Exotic Objects
[...]
whenever an own property of an Array object is created or changed, other properties are adjusted as necessary to maintain this invariant. Specifically, whenever an own property is added whose name is an array index, the value of the length property is changed, if necessary, to be one more than the numeric value of that array index;

By definition, an exotic object is any object that overrides internal methods that all normal objects have. In this case, the array exotic object overrides the internal [[DefineOwnProperty]] method so that when you define a property of the array such as setting an index, it takes extra steps to make sure things such as the property length is updated. Your Foo constructor does not create exotic objects, thus it doesn't override the [[DefineOwnProperty]] internal method like arrays do -- and thus doesn't update length when you directly define a value at an index.
Why does push push to the previous index?
Since Foos are not exotic objects and thus don't auto-increment length when an element is added directly with foo[1] = 'def', the length remains 1 when you try to push to it the second time. If we look at Array#push again:

22.1.3.17 Array.prototype.push ( ...items )
[...]


*Let len be ToLength(Get(O, "length")).

[...]


*Let items be a List whose elements are, in left to right order, the arguments that were passed to this function invocation.

[...]


*Repeat, while items is not empty
a. Remove the first element from items and let E be the value of the element.
b. Let setStatus be Set(O, ToString(len), E, true).


Thus, since your array length is still 1 because foo[1] = 'def' did not modify the length property, it would set the new to-be-pushed element at index 1 because the length was 1.
The same principle applies to Array#forEach. forEach depends on length to iterate through the array. Since your length is not modified when you do foo[1] = 'def' and remains 1, forEach only iterates from indices [0, 1) causing it to only log the first element. Pushing updates the length and causes it to iterate from [0, 2) and logs both elements.
Why shouldn't I do this?
It's because arrays are exotic objects. They do not get along with regular objects, in the sense that regular objects fundamentally cannot achieve the same behavior as exotic ones. Exotic objects, by definition, override the default behavior of internal methods to achieve certain behaviors needed for functionality. Arrays, in this case, have to especially handle setting indices and managing length -- done with the internal method [[DefineOwnProperty]]. With a regular object, it doesn't override [[DefineOwnProperty]] so many basic operations do not work correctly as a result -- so you shouldn't do it.
You could, though, use an exotic object such as the Proxy object to implement your own code for [[DefineOwnProperty]] like arrays to mock the behavior. Another way, as loganfsmyth mentioned, you could use ES2015 classes to extend and subclass builtin exotic object constructors correctly thus mimicking array behavior.

A: foo.push('abc');

foo.length; // = 1 as expected. But wait, why isn't foo.length undefined? 
            //How/when did this property get attached to foo? 

The .push() method is setting the .length, as it would with a real array. It isn't bothered that .length didn't already exist and it treated it as if it were 0 - so new element added at key 0 and .length set to 1.
foo[1] = 'def';

foo.length; // still = 1. But foo={0:'abc',1:'def'}, Why not =2?

Your object isn't an array, so although adding elements with numeric indices works (though they're converted to strings, as indeed real array object indices are) it doesn't magically set the .length for you like it would with a real array. The existing .length of 1 is thus unchanged.
foo.push('ghi');
foo.length; // = 2, and now foo = {0:'abc', 1:'ghi'}. So it overwrote the key=1,
// which means its accessing the same location, but the first approach
// did not change the length ( didn't become a part of the array ) why ?

The .push() method uses the existing value of .length to decide where to add the new element. According to .length you only had one element, so the new one was pushed in at index 1 and .length was changed to 2.

A: push() function updates the length of array so after foo[1] = 'def', length is still 1. foo is an object and hence assigning 'def' to key 1 works but doesn't increase the length and after foo.push('ghi') length gets incremented. Also, foo.push('ghi') uses current length of foo which is 1 at the time, as index for 'ghi'.
Also before you call the push() function for the first time foo.length is actually undefined. push() function defines it for the first time and after that increments it for each call.
