As I was sitting in my hotel room and, totally worn out, (and) perusing a tourist guidebook, there was a power outage.

If you want to include "and," it goes here.
"Tourist" is questionable. On the one hand, if you are in a hotel room exhausted and looking at a guidebook, the natural inference is that you don't live there, so that is an argument for deleting the word. On the other hand, if you want the reader to know that you are a tourist, then that is a good reason to keep it.

However, in less than a minute, I heard someone loudly screaming (loudly), and then the fire alarm went off.

If you want to include "loudly," it goes here.

I rapidquickly jumped out of my chair, grasped all the documents and money I had and dashed downstairs.

How are "rapidly" and "quickly" different? You can do the research, but "quickly" works better here in my opinion. If that word is too boring for you, I would offer "promptly" or "swiftly."

As I reached the reception desk, I saw a small group of people, my fellow hotel residents, talking to the administrator.

So, what you wrote is an appositive, as you probably know. However, it was misunderstood by at least one reader. That reader thought there were two groups, one small and one made up of other residents. To avoid this possible confusion, you could use a different structure. Again, as you may know, in English, appositives are widely used with names and titles, where their use is obvious.

In the meantime, the muffled cries haved become even more apparent, and it emergoccurred to me / it dawned ton me that they were coming from the elevator!

"Emerged" is not the best word here. Although thoughts and feelings can emerge, you can't use that word as a synonym for "occurred," which would work as a one word substitution. "It occurred to me" is similar to "I realized." For something more poetic, you could use "it dawned on me."

It turns out that a mother and daughter had been stuck in an elevator without power, and the father hasd triggered the alarm in order to notify the fire department.

You used the correct verb tense "had been," and you need it again "had triggered" for consistency.
I think "the father triggered" would also work, because the trapping came before the triggering.

Within roughly five minutes, the fire brigade arrived, and the family was successfully rescued.

I deleted "roughly" for two reasons. 1. It is such a short period of time, you can be more precise. "Roughly" is better used for estimates with a larger number, such as the distance between cities, the time to reach a distance place, etc. One could even make an analogy between a rough draft and a rough estimate. 2. it slows the sentence down.

While we were waiting for the firemen, the father told us that it was their first day in Paris.

A participial phrase at the beginning of the sentence is tricky. IF it is going to be used, you used it perfectly, by putting the subject immediately after the participial phrase. However, I think a simple structure is better here because it helps the reader grasp the situation more clearly. BTW "firemen" sounds a little outdated in English. Nowadays we tend to say "firefighters."

With tears and trembling hands, all three then went for a walk...

"all three then went for a walk" seems a bit of a let down after all the drama, which makes it a weak ending to such a well told story.
"Sallied forth" is much too upbeat; "embarked" is much too grand, but I think you can come up with an ending that is stronger.

It all happened on a sunny June afternooon in June.

As I was sitting in my hotel room and, totally worn out, and perusing a tourist guidebook, there was a power outage.

However, in less than a minute, I heard someone loudly screaming and then the fire alarm went off.

Screaming is already loud so there is no need to put loudly.

I rapidly jumped out of my chair, grasped all the documents and money I had and dashed downstairs.

As I reached the reception desk, I saw a small group of people, and my fellow hotel residents, talking to the administrator.

It turns out that a mother and her daughter had been stuck in an elevator without power, and the father has triggered the alarm in order to notify the fire department.

Within roughly five minutes, the brigadefirefighters arrived, and the family was successfully rescued.

As far as I know, brigade means a group of soldiers in the army.