The Gemara in Berachos does not say that if the person asking is an important person, then one should accept immediately. That is the ruling of the RE'AH, as cited in the SHULCHAN ARUCH (53:16). In response to your questions:

1. Yes, this applies today. However, as the Shulchan Aruch says, it only applies to someone who is not the regular, consistent Shali'ach Tzibur (such as a mourner during the first year, etc.). However, if one follows the advice of the Gemara as if it was a game, without understanding the point of refusing twice to Daven for the Tzibur (that is, to cultivate the trait of humility in oneself), then it would probably be best to just accept on the first request, since refusing is not a Halachic obligation but merely a recommended practice.

2. The Gabai is assumed to know this practice (that is why he is the Gabai), so he knows to ask the person two or three times before he accepts. Of course, being the Gabai is not always an easy job. When we learned Berachos, a Gabai sent us the question enclosed below.

3. As mentioned above, this practice applies only for non-regular Chazanim. It seems that it applies only to the central part of Tefilah -- the Shali'ach Tzibur who Davens for the Amud from Yishtabach until the end of the Tefilah, since that Shali'ach Tzibur plays the most important role. Also, in the times of the Gemara, it was this Shali'ach Tzibur who was actually recited Tefilah for everyone present to enable them to fulfill their obligation. Since his role is so important, it is necessary for him to cultivate the trait of humility when he is asked to fill that role.

4. An "important person," or in the words of the Poskim, an "Adam Gadol," is defined differently for different Halachos. Sometimes it is any person recognized as someone whom we are required to act towards with respect, such as a Talmid Chacham (perhaps even a wealthy person, according to the Gemara in Eruvin 86a).

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THE DAFYOMI DISCUSSION LIST

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Berachos 034a: Making oneself humble

Jeff Ram asked:

Question: Regarding the q & a on daf 34, your answer 2a, that a person should first

refuse, then hesitate, then go to the Amud follows Rashi, in d.h. y'sarav;

Rashi seems to say that he should be humble and pretend that he's not qualified,

etc. How is a person humble by "pretending" to do anything.

I was formerly a gabbai and often encountered men who, following the

suggestion of this gemara, would first refuse my invitation twice, and then

accept. However, I knew at the time (often by the way they refused, or by

the look on their face) that it was just a game, and I often wondered how

this could be the "humility" suggested by Rashi.

The Kollel replies:

Rashi does not say that one should pretend to be humble. He says, "Ya'aseh Atsmo," which, understood literally, means to make oneself as if he does not want to Daven for the Tzibur. That is, he should actually teach himself to act humbly so that he should become humble. (See Shulchan Aruch OC 53:16).

You are correct that if one follows the advice of the Gemara as if it was a game, without understanding the point of refusing twice to Daven for the Tzibur (that is, to cultivate the trait of humility in oneself), then it would probably be best to just accept on the first request, since refusing is not a Halachic obligation but merely a recommended practice.