As a Fizz Main since his release a few weeks before Season 2; I watched the video as well and thought he could use more clarity if anything for people playing against him. I don't think your changes really address his issues as a whole.
(For the purposes of this, fizz will have terrible health scaling, and start out with an amount of health similar to anivia's. But along with this, by level 18, have the lowest health in the game without buying items to compensate.)
This here makes playing him feel terrible. Being Melee means he has to endanger himself for CS and to be in range to make a move against an opponent. Lowering his overall Health and his Scaling Health will punish him farm more than he already gets in certain matchups ie: Morgana is a good example of this with her strong AoE wave clear and Q Binding.
Passive - Nimble Fighter: Whenever you dodge a targeted attack (either from being targeted by an ability and casting playful or hourglass or generally somehow don't get hit by it) or if an enemy skill shot comes near you without hitting you, you gain bonus movement speed for a few seconds. This does not stack but will refresh if it procs again. (Movement Speed scales with level)
As a whole, it's not a bad idea but I am betting this would be incredibly hard to code for on proc usage. How would you determine if you dodged a skill shot or the player just missed completely?. But regardless his passive is underwhelming and could be changed. Personally, I think Fizz's W passive would be a good application for his innate passive if given a little bit of a Darius treatment with an overhead visual. Like a Fish slowly being eaten for the duration with a total damage taken overhead.
Innate - Urchin Toxin: Fizz's Auto Attacks and Abilities rend his enemies, taking (1/2/3/4% of the target's missing health at Levels 1/6/11/16) stacking up to three times over a 3-second duration.
The reason I thought of this for his passive was because of how his current W has a lot of hidden power and the current implementation feels a little bad with the active portion of the skill.
Q - Urchin Strike: Fizz dashes a fixed distance through a selected target and deals a small amount of magic damage, This spell also deals both on hit effects, and on spell effects, if this kills an enemy minion the cooldown is halved.
(This spell scales with AP)
The Q change feels unnecessary wit how prevalent CDR is currently with the refreshed Mage items and being at max rank it's a 6-second cooldown the half refresh wouldn't be needed. So it can keep its current iteration.
W - Seastone Trident: Passive - Any time you deal damage, you deal a small percentage of a target's health as bonus magic damage, this damage is doubled if you currently have nimble fighter's movement speed bonus.
Active - Fizz will cartwheel on his trident clockwise or counterclockwise to the closest enemy champion within range, the direction depending on cursor location (sort of a counterclockwise dash? still targetable but you're basically rolling around your target). Afterwards his next auto attack deals all on-hit effects twice, including single instance on hit effects like spell blade and dead man's plate
You and I had the same idea about the DoT portion should be stacked which would honestly feel better from a gameplay perspective, but I have an issue with how the active works along with the double proc of on hit effects. It wouldn't feel great and there would still be a lot of hidden power in it without seeing it from the receiving end.
Instead, his W can keep its current active with no passive attached for stronger Magic Damage on Autos. (Maybe even apply a portion of it to turret damage?)
W - Seastone Trident: Fizz activates the Seastone in his Trident empowering his auto attacks with magic damage.
Cost: 40 Mana
Cooldown: 10 Seconds
Active: Fizz's basic attacks deal bonus magic damage for the next 6 seconds.
Bonus Magic Damage: 15/20/25/30/35 (+ 35% AP)
This way his W has a clear distinction of what does and you can see the damage from the active portion instead of half wondering how much is going to the DoT effect from the old W.
E - Playful / Trickster
Playful: Fizz hops onto his trident towards the target location, becoming untargetable for 0.75 seconds, before splashing onto the ground below him, dealing magic damage to nearby enemies and Slow icon slowing them for 2 seconds.
Cost: 90 / 100 / 110 / 120 / 130 Mana
Cooldown: 16/14/12/10/8 Seconds
Splash Radius: 270/330/400 (I think the wiki has it backward which doesn't make sense so I am guessing here.)
Magic Damage: 70 / 120 / 170 / 220 / 270 (+ 75% AP)
Slow: 40 / 45 / 50 / 55 / 60%
Trickster
SECOND CAST: Fizz immediately hops off his trident towards the target location, dealing the same damage to nearby enemies but without applying the slow.
Splash Radius: Is smaller but I don't know by how much sorry.
Magic Damage: 70 / 120 / 170 / 220 / 270 (+ 75% AP)
I took this from the League Wiki so if anything is off my apologies, but the reason why I posted it is because this ability is one of the most annoying and playmaking abilities in his kit and probably ranked in the top 3 playmaking overall in League.
With that said as in the video pointed out it's overloaded and probably slightly unbalanced, but overall it has many things tied to it. It has Fizz's only source of Waveclear and it's his only true escape ability. I won't repeat everything said in the video, but I personally don't want to see him lose this ability, I can understand if the damage was toned down by a small amount since Abyssal Scepter no longer lets him wave clear super easy. ( I really think Riot nerfed the item because of this one ability alone which is a bit funny to me, but it was a much-needed change.)
But on a personal not can we get a dash range indicator for this Ability? Most of the time you have to be way off the screen with the mouse to get it to correctly jump over the wall with either the first or second jump. It almosts feels bugged in how it works only half the time for wall jumping.
Now his Ult is pretty balanced overall, but I felt the change to it from all of Fizz's damage to just his Magic damage killed a playstyle which is still subsequently being played by melee mage assassins in the top lane it seems (Tank Akali comes to mind atm). So the idea I had was to just tweak his numbers a little.
Instead of "For the next 6 seconds, Fizz's magic damage against them is increased by 20% excluding Chum the Waters's damage".
Have it changed to "For the next 6 seconds, Fizz's damage against them is increased by 5/10/20% excluding Chum the Waters's damage".
This way it may allow AD Fizz to be viable again depending on the Meta and toning down some of his burst damage at the earlier levels.
This is just my thoughts on the Rework Idea if he is even being considered. And sorry for the large blocks of text.