Now the questions are:
1- Was there a moving firing pin in the block or was the block designed to fire the bullet ?
There was a separate firing pin in the block. I think the little cut you noted in the extractor is meant to cam it back into place without the need of a spring or extra part as found in nicer single shots.
2- Are there plans or detailed images of that block I can use to rebuild it ?
Not that I know of.
3- Is there a site or a book I can consult for that rifle ?
There has not been a lot of study of such a cheap little gun. I found one article and think there is another I have not yet tracked down in old Gun Digests.
4- What's the full name and model of my rifle ?
In 1901 Sears & Roebuck called it "Our $2.25 Remington System Flobert Rifle." They also said "Note: We do not recommend nor guarantee Flobert rifles. Buy a good rifle, it will pay in the end." US made Stevens rifles started at $2.95.
In 1911 Adolf Frank (Germany) called it the "Small Remington rifle, mod 1910, mark Crolet." Of course it is not a Remington, but a pale imitation. Crolet apparently the maker for ALFA sales and export.
5- Is there a site or a place I could possibly buy an old block for my rifle ?
Not that I know of. Even if you bought a junker in worse condition, there is little chance of the breechblock fitting yours. The two Floberts in the Gunderson list are of the same TYPE but with differences in detail. Made by different companies or at different time, parts not likely to be interchangeable.
The problem is, your rifle is over a hundred years old and of a cheap design and brand to start with. There is not much scope for repair. Finding a junker of the same series would be your only chance, and that not good.