The reason why the word "sumimasen" doesn't correspond exactly to English
phrases like "excuse me
", etc, is because of its literal meaning (duh).
One of the things that I don't like about teaching people set phrases in Japanese, is that they often don't mean exactly what we're told they mean. Arigatou gozaimasu is an example of this.
The word sumimasen comes from the verb sumu, to be ended, finished. Its use is based on the idea of reciprocation, that every act of magnamity should be rewarded with an action on your part. Thus the speaker who says sumimasen means that since you have done me this kindness, this matter is not finished between us.
So when you mean "sorry, I came late to class, forgive me", you say sumimasen, and when you mean "thank you for lending me your umbrella" you also say sumimasen. The net effect is to confuse foreigners.
And YES. Since sumimasen comes from the verb sumu, which can be written in kanji, sumimasen can, and regularly is, written in kanji: (in EUC)