I've made the MC pivot from pure mage into Spellsword, however the extra attacks don't seem to be happening at all.
I've tried with a staff, with a sword (crafted to have magic attack on it), I've tried on the front row and the back line. Targetted enemies on the front and back, just in case.

I have never seen a single target spell be followed by a Basic Attack ever, even though the description says it should happen 100% of the time. The combat log also doesn't show any attack or damage being done additionally after the spell. 

The same way, the active effect of casting a spell after a Single Target Physical Attack is also something I have never actually seen happen.
Does it simply not work or am I misunderstanding something about the mechanics? 

PS: Also, a bug, seemingly? I had Arcane Blade when I decided to Oblivion the MC and redo the classes a bit differently. It seems the Water Enchant I had active at the time lingered, turning all my physical attacks into Water. And also doing next to no damage. The damage seemingly dropping off a cliff is why I wanted to redo classes, I'll try and do testing with that when I get the class back because it seemed some interactions might've been tripping themselves into neutering the character. 

Do you play with mods? I assume you've activated the spell effect itself. Can you drop your save file?

Nah, no mods.
I think there's some in the mods folder but I don't really remember them and none are loaded.
If we're talking the Spellsword's spell effect, yes? I've toggled selecting spell on and off and made both physical skill and  basic melee attacks. 
But if we're talking following up a single target spell with a basic melee attack, I haven't seen something to activate.

Does the back/front row position matter for melee more than I'm aware?  Since the Spellsword's range is always melee now it might be it messed with something in a way I didn't consider.

For the Arcane Blade buff lingering through Oblivion I haven't tried yet, but I'm sure retoggling the element in the next battle MC is in should override/undo it. Those things seem to persist outside combat, so it's probably just that the game didn't account for losing the ability while it's active.

How do people usually share the save file? Thing was too big to fit a free pastebin. Do folks just paste it as a post here?

PS : As a side note, sometimes trying to open the console and untick Hide Errors to take a look kinda freezes the game. Is the stuff that would be written to console pushed to a log file that'd be easier to look at/share?

Well the easiest way is to drop it on discord or upload somewhere. Mod folder might affect things even if they aren't toggled. Judging from the errors it seems you might have lots of issues in it