Does this "Stock" Cooler come with pre-applied paste?

Yes.

Like a grey [strike]square.[/strike] or whatever shape this is:
My extra stock intel heatsink from my i5 4670k
lDY1PT7.jpg
 
Whadda mean installed?
I mean, it was already on the CPU; but I just took of the Cooler to check for paste, and I found it.
And I put the Cooler back ontop of the CPU.
Is it still alright?
 


1. Did you already put the CPU into the motherboard?
2. Was the cooler already on top of the cpu?
3. 99% of CPU coolers come with thermal paste either already applied or in a self applicable tube in the box.

If the CPU and cooler were already hooked up and plugged into the board, yes they will have thermal paste on both of them.

If a CPU doesn't have thermal paste applied between it and the cooler, it will experience dangerously high temps as there basically isn't anything besides air to transfer the heat from the cpu to the heatsink, and air transmit heat very very poorly.