Hey Natasha 
Great question!
Firstly let’s discuss some Sus chord theory:
A nice way to look at sus chords is a ‘half way point’ between the ii-7 and the V7 chord in a 251 progression.
In this case we are playing a V7sus chord instead of the ii-7 chord.
Looking at the chart in the iRealPro app, we can see that the chords for those 2 measures are:
E-7 → A7 → Bbmaj7.
Let’s play the notes of E-7:
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In your left hand play the root (E),
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In your right hand play the b3(G), the 5(B), and the b7(D).
Now if we replace the E root in our left hand with A low down in the bass, we now have A9sus. Our left hand is playing the root of A7, and our right hand is playing b7(G), 9(B) and the sus4(D)
One way to view the sus chord is simply the ii-7 chord, but with the root of the V7 chord in the bass. This is why I say it can be viewed as a “half way point” between the two chords.
Next let’s extend the sus chord up to the 13th:
Let’s now start with an E-9 chord and see what happens…
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again the root in our left hand (E)
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then play the b3(G), the 5(B), and the b7(D), and the 9(F#).
Again swap the root note for A, and we now have A13sus because the F# becomes the 13th of A7.
Then taking into account the melody:
The melody note is A, and so I simply stacked this on top and played the root and b7 in my left hand, and in my right hand I played the 9(B), sus4(D), 13(F#), and A(root). You could also look at this as a minor 7th chord off the 9, over the Root-b7 shell in the left hand.
This is a really handy voicing to remember and works great whenever you have the root in the melody over a dominant chord.
Resolving the tension
When playing a sus chord, it’s usually nice to resolve the suspended 4th down a half step to the major 3rd. In this case, the suspended 4th of A13sus (D) falls by half a step to become the major 3rd of A7 (C#) - also notice that I drop the 13 down a half step to become the b13/#5 - this adds some nice colour.
Again looking at the iRealPro Chart, we can see it says E-7:
As an alternate chord sequence, you could replace the A13sus with E-11 (because the 11 is in the melody) you could simply stack 3rds up sequentially from E to get the voicing, or alternatively, you could play the Kenny Barron Voicing if you hands will stretch that far.
Again it sounds nice. I do prefer the ‘floating’ and ‘mysterious’ quality created by that sus chord, but this should help you see how both the ii-7 chord, and the V7sus chord are interchangeable.
I’ve certainly digressed there but I think it is important information to understand 
Now Onto Your Questions:
In the lesson, we go from A13sus → A9b13 → D-7. This passage is functioning as a 251 into the iii-7 chord of the key, but instead of playing E-7 → A7 → D-7, we play A13sus in place of the E-7.
Notice on the iRealPro chord chart that the harmony is labelled as E-7 → A7 → Bbmaj7.
Here the Bbmaj7 is being substituted for the D-7. The Imaj7 and iii-7 chords share many of the same tones which makes them closely related from a harmonic standpoint.
In the “My Foolish Heart” example, I personally prefer the sound of resolving into the D-7, but also try resolving into the Imaj7 chord and follow your ears on what you like best.
Your next question:
The key difference is that a sus chord does not contain the major 3rd (sometimes it can have it in there, but generally it doesn’t).
On the contrary, an upper structure triad will always have the major 3rd in there and would never have the natural 4th, or suspended 4th. With dominant chords, if the 11th is included in the voicing, it is generally sharpened which removes that dissonant half step interval from the major 3rd and 4th.
By all means you could play that 9-#11-13 upper structure there, but when I play that, it sounds a little crowded for my taste:
Left hand shell (root and b7) which is A and G
Right hand plays 9-3-#11-13-root which is B-C#-D#-F#-A (totally possible to play but it is quite a handful 
)
Here’s another option:
Instead of A13sus, play E-11 which would be E-G-B in left hand, and D-F#-A in the right hand.
Then go to a rootless A7#5#9:
left hand plays b7(G) and major 3(C#) and the right plays an F major triad in 2nd inversion with A on top which is the melody. Your hand should slightly overlap when playing this voicing, your right hand thumb is playing C, and you left hand is playing C#.
From top to bottom, we have left hand: G, C# and then right hand C-F-A.
Then resolve to D-7.
That sounds really nice to my ears 
From reading the last half of your question Natasha, i think you would benefit from watching these lessons to get a deeper understanding of sus chords and dominant chords.
Firstly the theory lesson on “Understanding Sus Chords”
This lesson will help you understand the constructionn, function, and application of sus chords.
And here’s 3 jazz standard lessons where we specifically work on sus chords and apply them as rehamonisations. In order of difficulty:
1) “What Are You Doing The Rest Of Your Life” (3 part lesson)
2) “The Shadow Of Your Smile” (2 part lesson)
3) “I Fall In Love To Easily” (some advanced applications here):
They are 3 of my favourite tunes… I hope you enjoy playing them as much as I do 
