Cuando pienso en los cactus, siempre me vienen a la cabeza los desiertos.

"Me vienen" refers to yourself or something with you, instead "siempre vienen" is like "they always come". For example:
*Siempre vienen temprano = They always come early
*Siempre me vienen a ver = They always come to see me
*Siempre vienen cansados = They always come tired
*Siempre me vienen a dar regalos = They always come to give me gifts

Sin embargo, aunque es verdad que se pueden encontrar en los desiertos, la realidad es una poca más compleja.

You can use "se pueden" but in other context:
*Se pueden encontrar en el bosque = Can be found in the forest
*Se pueden cocinar = Can be cooked
*Se pueden comer = Can be eaten
*¿Se pueden ir? = They can go?/Can you go?

In this case "que" is necessary to refers something probable: "que se pueden":
*¿Es cierto que se pueden quedar? = It's true that you can stay here?
*Sabía que se pueden cocinar = I knew that they can be cooked
*Ella no sabía que se pueden comer = She didn't know that they can be eaten

PEn primeramente lugar, un "desierto" es realmente una categoría muy diversa.

"Primeramente" is correct, but doesn't sound so natural. "Primero" (first) or "En primer lugar" (In first place) it's better.

Por ejemplo, ela Antárticoa se clasifica como un desierto, pero dudo que haya un cactus ahí.

"Antártica" refers to a place, "Antártico" refers to something from Antártica.
Antártica is a female word, you need to use "la".

"Pero dudo que hayan cactus ahí" -> "Hayan" is plural from "Hay", refers to cactus in plural
"Pero dudo que haya un cactus ahí" -> "Haya" is singular + "un" (indicates one element) refers to cactus in singular

Lo segundo es que no hay solo una especie de cactus, sino más de dos cientos tipos distintos.

One hundred = 100 = Cien (Long explanation)
Two hundred = 200 = Doscientos (dos (2) + cientos)
Three hundred = 300 = Trescientos (tres (3) + cientos)
Four hundred = 400 = Cuatrocientos (cuatro (4) + cientos)
Five hundred = 500 = Quinientos (Long explanation, down from this)
Six hundred = 600 = Seiscientos (seis (6) + cientos)
Seven hundred = 700 = Setecientos (Long explanation again)
Eight hundred = 800 = Ochocientos (ocho (8) + cientos)
Nine hundred = 900 = Novecientos (Long explanation)

So, let's start with "Cien". It's an apocope of "Ciento", both come from latin "Centum" that refer to one hundred of something. A similar homophone word is "Sien" it refers to the temple (two opposite sides of your head).

Now, "Quinientos". It comes from "quingentos" from the numeral latin "quingenti-ae-a" composed by "quinque" (5) and "gent-i/cent-i" that refer to "Centum". [http://etimologias.dechile.net/?quinientos#:~:text=El%20adjetivo%20numeral%20quinientos%20viene,ae%2Da%20(quinientos).&text=Esta%20es%20la%20palabra%20que,%2C%20diezmo%2C%20diciembre%2C%20etc.]

And finally, "Setecientos". "Sete" comes from latin "Septem" that refer to the number seven, and "Cientos" is a plural of "Ciento" from "Centum".

The origins of Spanish cames from latin. It's useful to understand why many words are the way they are.


Although "Sólo" means "Only" and "Solo" means "Alone", the accent mark isn't so relevant, people use "Solo" for both things. Personally I still using "Sólo", but is a personal preference.

Además, son nativos de muchos lugares diferentes, desde las Américas hasta África y Asia.

América is a continent, a singular word as "África" and "Asia". "Las Américas" is in plural.