Do I have to put on a "skate" to use my tango shoes with heels outside?

Do you have to put "a skate" (or a rubber sole) under the heeled shoes to use them "in the city", so other than for dancing?

Good news, the answer is "no"!

The outsole of your Lalatango is made of leather. It is a bovine leather called "croupon" (cut in the back of the animal) which gives a solid and resistant sole. And waterproof!
The leather sole gradually adapts to the shape of the foot, which makes the shoe more comfortable and promotes natural walking. It is perfectly usable in the city as well as at the dance.
It's the same sole found in many of the major high-end brands of dress shoes.

It is not necessary to put on a skate, in fact we rather advise against it because your shoes would lose their flexibility which also contributes to comfort. Only do this if you expect heavy use on rough ground.
The "big high-end brands of dress shoes" give exactly the same opinion.

But if it rains, will the leather sole hold up?

The sole leather is thicker and stiffer than that used for the upper of the shoe (= upper). Its production is different. Its tanning is longer (the leather is then immersed in different baths of aqueous solutions... so it macerates for a long time in water!), it is then compressed.
In short, the sole is also water resistant. Even if it is better to avoid making kilometers in the rain. And remember to dry your shoes well when they have been wet.

Please note that things are different with flat dance shoes whose ground sole is split leather (suede) , which is not suitable for outdoor use . In this case, the information appears clearly on the page of this site describing the pair of shoes.