Technically excretion does lower the total thermal energy of the body as it is leaving your body with the excretion, but it does not really in itself cool down the temperature of the body as the temperature is the same as your body (unless you define cooling down as negative change in the thermal energy needed to remove to lower object's temperature to zero K). Though if for example you drink cold water it does cool your body very slightly and urination could be thought of as dumping of used coolant.

Excretion does remove thermal energy from the body, but it also removes mass, and as a result your heat to mass ratio doesn't change in any meaningful way.

"Keeping it in" would technically make your body temperature slower to change (Up or down) because there is slightly more mass to heat or cool. Excreting would technically make your body temperature slightly more susceptible to change, again because there is less mass to heat or cool. But really, those changes are inconsequential.

The actual cooling would occur on intake, not excretion. When you drink cold water, your body heat will dissipate into that water until the temperatures match, resulting in a slight reduction in temperature.

So in summary, excretion itself does nothing to cool you down, even though it's taking thermal energy away, but the entire cycle of drinking cold water, heating it in the body, and then excreting it would reduce body temperature ever so slightly.

I'm not sure about number 2, but your bladder actually acts as a heat sink since piss is mostly water. Water has a crazy high heat capacity, so it drains heat from your body. It's a common tip for camping in extreme cold temperatures to not go to bed with a full bladder since your body has to spend some amount of calories warming up the piss.

When you say "cool the body" do you mean decrease the average temperature, decreasing the total thermal energy, or decreasing the total potential thermal energy? For the last two options the answer is a pretty clear yes because, even though our bodies try to efficiently consume food, poop still does have a bunch of calories in it and none of our waste leaves the body at 0K.

Is your poo requesting political asylum?

The heat capacity of the human body is around 3.0kJ/kg°C, and the one of urine is likely close to water (4.2kJ/kg°C). As such when urinating a person loses proportionally way heat than mass.

So if you're feeling hot perhaps relieving yourself might give you a bit more comfort. Although you'll likely do it anyway, as you drink more water.

For faeces it's trickier, as their heat capacity varies quite a bit depending on the moisture content, from 4.6 to 0.4kJ/kg°C. I'd guess that for healthy individuals it should be the same effect but in smaller degree as urine.

~~(And thus the No Poop Challenge becomes acclaimed as the solution for energy waste.)~~

Farting does expel some amount of body heat.

Im sure long-distance runners might relieve themselves without stopping, as it cools them down while also a release and embarrassment which might spur their brain into flight response.

I doubt if the heat loss is significant but when actively working out or doing intense work the mind might get tricked into thinking that it benefitted.

Anecdotally when running a fever it does feel better emptying out. So not sure if physiological or psychological effect.