Uses of 'すみません'

Question

I came across many situations where the word 'sumimasen' was used:

1. Apology
2. Thanks and apology
3. Making a request
4. Getting attention
5. Taking leave

Above are most common I could find, however, take following examples:

1. Imagine that you are at the bank, sitting with a group of people waiting to be served at the counter. Suddenly, the man next to you is called. “Ieda-san! Omataseitashimashita (thank you for waiting)!” the clerk shouts, and the man responds with “sumimasen.”
2. A woman goes to the department store to buy a new bicycle which requires her to fill out a registration card. After she fills it out, and the clerk assisting her checks it, the clerk returns it saying “sumimasen,” and of course, she replies, “sumimasen.”

At these two situations, I was very much confused as why 'Sumimasen' was used. Could you please enlighten me up?

Answer 1

Using すみません is very important in Japanese and as you've noted, it is used in a lot of contexts. (There have also been many questions here about it).

A few of the relevant ones to look at as references:

What is the difference between すみません and すみませんでした?

Using すみません instead of ありがとう

Usage of すみません (sumimasen) versus ごめんなさい (gomen'nasai)

Is there a difference between すみません (sumimasen) and すいません (suimasen?)

Etymological Definition

すみません is the polite negated form of すむ (済む). 済む means to "complete or end something." At least to me it was helpful to learn this, because etymologically what すみません or すまない (it's less polite cousin) mean is "I was inadequate" or "I was unable to complete [x] / do [x]"

Functional Definition

JDICT translates it as

excuse me, pardon me, I'm sorry.

Goo's dictionary defines it as

 The etymology of being the polite negation of 済む *plus* a phrase     
 that is employed to express thanks, imposition of a task, or that  
 I've done something wrong.

This is not exactly identical to the English (or may be just my American English?) "I'm sorry" as in "I feel emotionally terrible for having done this" and "I regret having done this" in that in Japanese this phrase is used all over the place for all sorts of things for which I have zero feelings at all.

Application

The two examples you picked don't seem like the clearest ways to understand すみません but the phrase is used a lot every day.

For the bank example, my sense as to what's happening is Ieda-san is apologizing that he's about to make the clerk do work for him (or alternatively that he's so blandly unidentifiable that the clerk has to yell for his name rather than knowing who he is). Alternately, he is apologizing for not having been able to take care of the matter himself and thus forcing other customers to wait.

For the department store example, I would imagine the clerk is apologizing for making her do something, and she in turn is apologizing for making him do something.

1. Apology - I am sorry I could not accomplish this.
2. Thanks and apology - I am sorry I could not accomplish this [if you want to be thankful add ありがとうございます]
3. Making a request - "I can't accomplish this on my own so ..."
4. Getting attention - "I am sorry that I can't do this on my own and have had to get your attention"
5. Taking leave - "due to my inadequacy, I can't stay until the end of what is going on"

Answer 2

In the end it all boils down to "sorry".

Clerk feels sorry to bother you with his request to fill the form.
The client feels sorry to bother the clerk by having him review her form and assisting her.

When you use it to say thank you, it is still essentially saying sorry for having you do such a thing for me.

When you use it to take a leave, it is basically saying sorry to everyone else for being so rude as to leave before everyone else.

When the old man responds to the clerk, it was probably meant as, sorry for having you go through the trouble of calling me, or sorry for having to deal with the request I am about to make.