Adding more thermal paste to stock cooler

No, adding more would be bad. What you want to do is to clean off all existing paste (use isopropyl alcohol if you can) and apply a bit of fresh paste.

If you're talking about the layer that comes with a new cooler, you can usually use it directly - you don't need to use your own at all. Replacing paste is only really worthwhile for old paste.

Also, the correct saying here is "less is more"¹ :)

With thermal transfer from the integrated head spreader (IHS, the metal on top of the CPU die) to the heatsink, it vaguely goes:

• metal-to-metal contact: best
• metal-paste-metal contact: alright
• metal-air-metal contact: very poor

So your best-case scenario is if you can maximise direct metal contact between the IHS and the heatsink. That means they should be as clean and smooth as possible, and a fair amount of pressure pushing them together.

Now, if direct metal contact is best, why do we have paste? Because it's very difficult to get solid metal smooth enough for perfect contact, so you inevitably end up with lots of tiny air bubbles, resulting in poor transfer. Adding paste fills up these little gaps, but adding too much paste¹ will either form a thick layer and prevent direct contact, or will end up getting squished out the side.

Even worse is trying to apply fresh paste on top of existing old/dried paste - that way you have the poor performance of dried paste (which can no longer spread effectively once disturbed) plus an additional layer. It's much better to just clean off the existing gunk first.

¹ You'd want enough paste. There's a bit of leeway here, but you also don't want to go squeezing a whole tube in - once you have enough, adding more won't help. Keep in mind that what looks like a tiny bit will actually spread out quite far once pressure is applied - you're squeezing a 3mm-high blob into less than a tenth that height. Optimally, you'd have somewhere maybe a little bit over enough.

For those interested, there's further discussion of specific application techniques and their relative effectiveness here: https://www.pugetsystems.com/labs/articles/Thermal-Paste-Application-Techniques-170/

Absolutely not. Thermal paste should be just enough to fill any gaps. A thicker than required layer of thermal paste reduces the efficiency of the paste. It is also not a good idea to mix different thermal pastes unless you know they are chemically compatible. Additives in one paste may break down additives in the other, producing compounds that may degrade the paste.

There's a reason heatsinks aren't made from enormous formed blobs of thermal compound. The BEST thermal paste is about 8W/m^2*K. Even steel is about 6 times better at conducting heat, at 50W/m^2*K. Aluminum is 205. They're not even close -- use the least amount of paste you can to fill the air gaps (air being 0.024W/m^2*K). You can actually skip thermal compound entirely if you lap the heatsink and CPU to a mirror polish -- high end overclockers do this occasionally.

You should apply enough paste so that when you put the cooler on, a small amount of paste appears on the sides. Putting less paste means you're still having air gaps which were not completely filled. Putting more means you're wasting paste, and if you apply too much it may spill on the board and you'll have to clean that.

Of course, you should completely remove the old paste before you apply the new one. The old paste has probably dried up compared to a fresh one, and probably accumulated some dirt and dust. This will prevent it from flowing well under pressure, and you will end up with more air cavities as a result.