Mi prenas ombrelon, kiam estas vento kaj pluvo.

Because ombrelon is the direct object of prenas.

In Esperanto, the ending -n marks the thing directly affected by the verb. Here:

Mi = the person doing the action
prenas = take
ombrelon = the thing being taken

So ombrelo becomes ombrelon because it is what the speaker takes.

Esperanto does not need a dummy subject like English it in sentences about weather or general conditions.

English says things like:

it is raining
it is windy

But in Esperanto, you can simply say:

pluvas = it is raining
estas vento = there is wind / it is windy

So kiam estas vento kaj pluvo is normal Esperanto. There is no need to insert a subject just because English would.

Both styles are possible, but they are slightly different in form.

estas vento kaj pluvo literally means something like there is wind and rain. It uses the nouns vento and pluvo.

A more verbal version could be:

kiam ventas kaj pluvas

That means when it is windy and raining.

The sentence you were given chooses the noun-based expression. It is completely natural and easy to understand.

Because kiam means when, and in this sentence the idea is whenever that situation happens, not if it might happen.

kiam is used for something seen as a real or regular occurrence:
I take an umbrella when there is wind and rain.

se would make it more conditional:
I take an umbrella if there is wind and rain.

In many situations the two can be close in meaning, but kiam fits well when talking about a habitual action tied to a situation that occurs.

In Esperanto, the present tense often expresses a habitual or general action, just like English I take can.

So Mi prenas ombrelon, kiam estas vento kaj pluvo means that this is what the speaker usually does in that situation.

It does not have to mean only I am taking an umbrella right now. Context tells you whether the present tense is:

right now
or
generally / habitually

If you wanted a future meaning, you could say Mi prenos...

Because la is only used when the noun is specific and identifiable.

Here, ombrelon just means an umbrella, not a particular umbrella already known to both speaker and listener. Esperanto has no separate word for English a/an, so the simple noun often carries that meaning.

Likewise, vento and pluvo are being mentioned generally as conditions, not as specific, previously identified things.

So the absence of la is normal.

Yes. That is also correct.

Esperanto allows either order:

Mi prenas ombrelon, kiam estas vento kaj pluvo.
Kiam estas vento kaj pluvo, mi prenas ombrelon.

The second version puts the time/condition first, which may sound a little more like English word order in many cases. Both are natural.

A comma before a subordinate clause is very common and recommended in Esperanto.

Here, kiam estas vento kaj pluvo is a subordinate clause, so the comma helps show the structure clearly:

Mi prenas ombrelon, kiam estas vento kaj pluvo.

If the subordinate clause comes first, the comma usually appears after it:

Kiam estas vento kaj pluvo, mi prenas ombrelon.

So the comma here is standard punctuation.

In this sentence, preni is being used in the natural sense of take an umbrella with you.

English often says take an umbrella, and Esperanto can do the same with preni ombrelon. The idea is that you take it along because of the weather.

If you wanted to make the idea of taking along even more explicit, Esperanto can also use kunpreni, but prenas ombrelon is perfectly understandable and normal here.