Learn Hardcore Japanese: eki de kasa wo wasurete simaimasita. - I accidentally left my umbrella at the station.

The particle で indicates the location where an action takes place. Since the act of forgetting (忘れる) happens at the station, you say 駅で. Using に would suggest a point of arrival or destination, not the setting of an action, so 駅に would sound odd here.

What role does を play in 傘を忘れてしまいました?

Particle を marks the direct object of a transitive verb. Here, 傘 is what is being forgotten, so it takes を to show it is the object of 忘れる.

What nuance does ～てしまいました add compared to just 忘れました?

The construction verb-stem + てしまう expresses that something is completed with a nuance of regret, accident, or unintended result. In polite past form (てしまいました), it means “unfortunately ended up doing X.” So 忘れてしまいました conveys “I ended up forgetting (and I regret it).”

How would you say this sentence more casually in spoken Japanese?

In informal speech, てしまいました often contracts to ちゃった. So you could say:
駅で傘を忘れちゃった。

Could you use a different verb to express “leaving something behind” instead of 忘れる?

Yes. 置き忘れる means “to leave behind accidentally.” So you could say:
駅で傘を置き忘れました。
This emphasizes you left it behind by accident, but the overall meaning is very similar.

What is the reading of this sentence in hiragana?

えきで　かさを　わすれてしまいました。

What level of politeness is used here, and how would you make it plain?

Using ～ました makes the sentence polite. To make it plain (casual), you’d drop ～ました and use plain past:
駅で傘を忘れた。

Why not say 駅で傘を忘れた if I just want to mean “I forgot my umbrella at the station”?

You can say 駅で傘を忘れた, but it simply states the fact without any nuance of regret and is in plain form. 忘れてしまいました adds politeness and expresses that you’re sorry or embarrassed about forgetting it.