League of Legends (LoL) Question: Does Teemo's poison ticks proc electricute

Teemo poison ticks do not proc electrocute, it does not count as an adition spell so things like Conqueror and alike do also not add aditional stacks. The only rune with stacks that synergizes with poison is Grasp of the undying as it keeps resetting the cooldown and the stacks (Meaning you have more time to hit enemies with grasp). Teemo sometimes uses electrocute to have more burst, since Q + mushroom + auto attack procs it quickly and just auto attacks do as well.

If you don't understand something let me know.

Yes if he use his 1st or Ult also

Electrocute does not proc on the poison.
There were times when RIOT messed up with the code, and it did count as a "Tick".

Teemo's Poison can, however, prolong the effect of the duration for Electrocute.
One auto...wait...wait...wait...
Auto again...wait...wait...
Auto for the final time, procing electrocute.

It's good for prolonging the pokes to juke the enemy stuck under tower when they expect only one auto and no burst when on low HP trying to grab that cannon minion.

As far as I know, No, it stacks per attack or cast, so if you refresh the poison tick with 3 separate auto attacks only then will it trigger electrocute. If you build Nashor's Tooth you should be able to sneak in 1 Q and 2 autos.