SECTION 1
B Reverse changes direction, but does not change momentum, and it is performed by completing the control stick input (i.e. the input itself as well as returning the stick to neutral) and then inputting the Special input. For example, when Snake is moving to the right, completing a left control stick input followed by a neutral special input will reverse the direction Snake is facing (so that he is now facing left), retain his rightward momentum, and initiate the Grenade special.
Recoil Special (also mistakenly called a B-Reverse, also correctly called a wavebounce) does not change direction, but does change momentum, and it is performed by completing the special input and then inputting the control stick input immediately afterward. For example, when Snake is moving to the right, completing your side B special followed by a left control stick input will cause Snake to recoil special, meaning he will maintain his direction, but his momentum will be reversed to a leftward momentum.
Essentially, you're doing the control stick input at the exact wrong time.
Opposite Control Stick input before the Special will result in a B Reverse.
Opposite Control Stick input at the same time as the Special will result in a B Reverse Side-B
Opposite Control Stick input after the Special will result in a Recoil Special / Wavebounce.
Additionally, the B Reverse seems to me to be far more lenient on the timing because you can tap the opposite direction long before the special input, and so long as there is no other control stick input between the two, it should reverse your special. I'd have to test this more extensively, but it's not exactly critical when you can just do it fast. The Wavebounce has to be done very quickly after the special, so it's honestly pretty phenomenal that you're able to do it so consistently, especially with neutral B (only upB is a harder input IMO).
SECTION 2
The 1:29 example does look a little odd to me, so I'll play around with this a little bit (I need to practice wavebounces anyway now that I've figured out what's going on with them!) and get back to you.
I've got something else wrong here I guess, because that 1:29 thing is also a Recoil special, but it's also a B-reverse, so there's some way to do both simultaneously. I can do this with UpB somewhat consistently by doing a Half-circle turn, starting with my dash direction, then the up direction and then turning to the direction I want to travel with up b. I'm unsure of how to transfer that ability to the neutral b, though.
Using B-sticking, I came up with this:
XXXXX//Left Control | Right Control || Left to Right ||| Right to left
Left C-Stick//Normal Side-B | Wavebounce || Reverse + Wave |||B-Reverse
Right C-Stick//Wavebounce | Normal Side-B || B-Reverse ||| Reverse + Wave
So: given that the left/right C-stick will be the special input, your control stick vs. your special inputs relate like this:
The Same: Side-B
Different: Wavebounce
From Same to Different: Reverse + Wavebounce
From Different to Same: Reverse
As for Grenades (and neutral special moves in general) I can only guess that Prof (knowing he doesn't B-Stick) uses the non-intuitive direction-detection of the game's engine to input a direction while not inputting a side special. To test this, tilt your control stick very slightly so that Snake starts walking. If you get a side-b, work it very slowly back to center until you start getting a grenade. It's just outside the 45 degree angles (which result in side-b). You can compare this to tilts, which require a far smaller X axis change to perform.
This is getting really technical, stop me please.
Also would not endorse the use of a Special c-stick, especially if you're really accustomed to the smash c-stick, because you cannot throw grenades with b-stick, on top of not being able to do retreating aerials, quick aerials oos, SHFFLs, et al. This is a really tight input window, but if you just needed to face the opposite direction while moving toward the stage will recovering, the very first section of this post will help you do that. If you want to do what Prof is doing at 1:29, you'll need to sit down and tinker with this mechanic for a good hour, 'cause I can't get it very well at all in my 15 minutes of free time.