What temperature should eggs be when I beat them?

Assuming you're talking about beating egg whites, the best results are obtained with fresh, cold eggs (as you noticed).

Warmer (or older) eggs will gain volume quicker, but the resulting foam will be less stable. Since nowadays everybody will likely beat the eggs using some form of mechanical mixer, the time factor is now irrelevant, while the foam stability appreciated.

The "baking rule" probably originated when refrigeration and mixers weren't around.

Here's a list of things influencing max volume and stability:

Age: as eggs age, their pH increases: around 7.6 when laid, 9.2 after just three days. Lower pH is better for stability.

Temperature: temperature at the time of beating does not really make a difference, but if you refrigerate the eggs you'll delay protein denaturation, increasing stability. Heating the white to 58 °C for three minutes increases the foam formation ability, but over that protein denaturation will ruin everything.

pH: adding a bit of acid (citric, tartaric) will help both volume and stability. Citric acid can also help with keeping the foam white.

Water: adding water (up to 40% of weight) will yield a less dense, slightly less stable foam

Sugar: adding it will slow foam formation, and yield a denser, stabler foam.

Yolks and fats: Avoid. A single drop of yolk can reduce the final volume by two thirds.

Copper bowl: lengthens the time needed, but will yield a stabler foam.

Salt: shortens the time need, but will yield a less stable foam and increases syneresis (weeping).

On a side note, there also is a myth that older eggs are much better for macarons, but it has been debunked.

The higher the temperature is, the better for beating. In fact, some applications which require the eggs to really hold their foam well, like mayonnaise, are very hard to get done with cold eggs, so are made with eggs in the 50 - 85 Celsius range, frequently over a water bath. If you are talking about pure yolk, I seem to recall that 72 C is the optimal temperature, I don't remember coming across such a number for whole eggs.

Also, the recommendation is not simply about beating. Egg coagulation is pretty complicated stuff and depends not only on the final temperature your egg reaches, but also on the speed at which you warm it up. Then there are other side effects like not wanthing to mix melted butter or chocolate with cold stuff. Your results are always smoother if you started baking with room temperature eggs than with fridge temperature eggs (although the difference can be too slight to notice in some coarser applications like pound cake).

I don't know why your eggs seem to beat up better when cold. One reason could be some kind of observation bias. Another - I am speculating here a bit - is that your eggs age quicker if kept on the warm counter, so if you buy them and hold them around for a week, I can imagine that the less-aged eggs from the fridge create a nicer foam than the half-liquefied eggs kept at room temperature. If your environment is so hot that this makes a difference, try baking with fresh eggs only, or store in the fridge and take them out 24 to 48 hours before baking.