First of all, sorry for my english. I'm not native.

Napoléon's succession always was for him a source of enormous troubles.

Soon after Napoléon self-proclamed Emperor, he really soonly faced a huge problem of his descent, because Napoléon's family was not extremely noble and pretty far from french land (they were a Corse rich family, and they hated Napoléon's wife, Joséphine de Beauharnais)

Napoléon had a wife, Joséphine de Beauharnais, which already had 2 childrens before getting married with Napoléon. For Napoléon, succession was, like kings in France, based on descent and only males.

The huge problem that faced Napélon was...that he just could not have kids with Joséphine. Today again, nobody knows why. At first, we suspected Napoléon of being sterile because Joséphine had had 2 childs already.

At this point, Napoléon, despesperate, had an idea to continue his descent. He planned to adopt the son of Joséphine de Beauharnais' daughter, Napoléon-Charles Bonaparte. Basicaly, his step-Grandson's. And it was OK for the descent because, as written in the Imperial constitution

" Napoléon Bonaparte peut adopter les enfants ou petits-enfants de ses frères, pourvu qu'ils aient atteint l'âge de dix-huit ans accomplis, et que lui-même n'ait point d'enfants mâles au moment de l'adoption. - Ses fils adoptifs entrent dans la ligne de sa descendance directe. - Si, postérieurement à l'adoption, il lui survient des enfants mâles, ses fils adoptifs ne peuvent être appelés qu'après les descendants naturels et légitimes. - L'adoption est interdite aux successeurs de Napoléon Bonaparte et à leurs descendants."

which basically say that: "Napoléon can adopt the childs or grandchildrens of his relatives after 18 years old if he still doesn't have childs and thus made them pretenders to the throne at his death."

Napoléon truly loved this child, he could adopt some others childs, like Joséphine's first son, or even in his own blood family, but no, he just...liked him.

The sources of the time paint a totally different Emperor with this child. Napoléon, the respectfull and severe Emperor, loved this child so much that he was spending pratically all of his time with him. One anecdote that reveal well that love he had with him was, as written in a letter from the Empress's daughter Hortense de Beauharnais (Charles-Napoléon's mother), that Napoléon was litterally lying on his knees on the floor to play with this little kid. It was so unusual that a quick drawing of this scene was made with a pencil, and it's pretty much self-explanatory (I unfortunately could not find it on google, saw it in a museum). This child died in 1807 of measles, and Napoléon was so affected by it that he organised a contest of the best thesis on how to cure measles.

But the event that change everything came. During his campain, Napoléon had, indeed, a child with this Polish woman, Marie Walewska, in 1806.

To respond to your first part of your question, yes, a lot of people knew about it, certainly not all the farmers in the countryside, but, because this child was a very important child for him (i'll explain just after), it was well known that he had 2 unlegitimate childs.

This illegitimate child, Léon, born in 1806, was very important for the emperor, because it was the proof that Napoléon was not sterile. Thus, people started suspecting Madame de Beauharnais to have becoming sterile with age (she was at this time 43 years old and nowadays, historians suspect her to had had an early menopause due to an infection or a bad fall).

At this point, the wheel turned for Napoléon, which the fact that he couldn't had a baby was a source for him of a enourmous shame, and a "self-disgrace" (he wrote it in some of his letters for her wife during the 1799-1806) was now not his fault.

And, thus, Napoléon, was forced to divorce to his wife Joséphine de Beauharnais in 1810 in front of the impossibility to have a child with her. This event was so important for Napoléon that the 2 letters of divorce, the one from Joséphine and the other from Napoléon, are truly delightful to read. Napoléon and Joséphine loved each others, truly, even if the relation had high and down (this need and other askhistorian thread) and this forced separation, for the good of the Empire, in fact, killed Joséphine which became depressive.

Here are some extracts from Joséphine divorce letter that are pretty explanatory: " je dois déclarer que ne conservant aucun espoir d'avoir des enfants qui puissent satisfaire les besoins de sa politique et l'intérêt de la France, je me plais à lui donner la plus grande preuve d'attachement et de dévouement qui ait jamais été donnée sur la terre."

which can be translated by "I have to declare that, conserving no hope to have childrens that can satisfied the needs of France interests and France politics, i'm pleased to give him [Napoléon] the greatest proof of attachement and service that had never be done in this Earth" (she talks about divorcing)

And another one: "mais la dissolution de mon mariage ne changera rien aux sentiments de mon cœur : l'empereur aura toujours en moi sa meilleure amie"

which is "But, the dissolution of this marriage will never change nothing to our feelings in my heart, the Emperor will always have with me his truly best friend"

Anyway, Napoléon then had marriage in 1810 with Marie-Louise d'Autriche and had this children, Napoléon II, who, by the way, became mecanically emperor of French from the 22nd of June 1815 to the 8th of July 1815.

Problem is, this children had no support from nowhere and was not even in France during his "reign", and had only 4 years old... Very soon, he was totally evicted from throne by the royalists (Louis XVIII, king from 1815-1824) that just didn't even dedicated one minute of attention to him. He died in 1832.

To finish with those 2 unlegitimate childrens and to answer some part of your question. He indeed had them in Poland, and because he was not married with thoses womens, those childs were a bit of a shame and just could not be in any kind of form pretendant to be Emperor. Also, to speak about there claim over Napoléon III, what is interesting is that Napoléon never hid thoses childrens. The first to be born surely tried to claim the throne, but was soon evicted from it because the general opinion was cleary in favor of a "french legitimate emperor" and not a "polish bastard", he however succeeded to get some titles of nobility by dint of insisting (and it was a great occasion to make him quiet and to escape to potentials future problems).

For the one born in 1810, Alexandre Colonna Wlewski, he just never claimed the thrones, his named didn't even had Bonapart in it! Althought he was a really close friend to Napoléon, he even visited him on the Island of Elbe during the first exile of Napoléon in 1814, and was decorated with a lot of military distinctions.

please indicate me any English mistakes if you see some (it should not be difficult to see some)

Sources: Secret d'Histoire: Joséphine de Beauharnais, Imperatrice, 2018

Letters between Napoléon and his Wife

"Napoléon's letters", 1934

Museums and lots of self reading

Archives of documents involving Napoléon's marriage that I read at the National French Library

Mémoires intimes de Napoléon 1er

Wikimédia for some scanned archives

EDIT: As this is interesting some people, here are some extracts I loved that Napoléon sent to Joséphine. they can be found there in french http://www.bmlisieux.com/curiosa/napoleon.htm

" Albenga, le 18 germinal, an IV, (7 april 1796)

I have never spent one night without loving you, i have never spent one night without you in my arms. I have never drank one tea without sermoning the glory and the ambition that is making me far from you, the soul of my life.

At the middle of my troops, crossing the fields, my adorable Joséphine is alone in my heart, in my mind, you absorb my conscience. "

Then....8 days letters...

"Ah !! Malicious !! How could you write a letter like that to me ?! She is so cold ! Furthermore, for 4 days you didn't wrote to me ! [...] My soul is sad, my heart is slave, and my imagnation is scaring me...because you don't like me anymore, one day, you'll hate me, just tell me it."

Then... 30 days later

"Ah! My good friend...I admit...I was wrong. During spring, countryside is beautiful, like you. I know you've been unfaithfull with this 19 years old boy [...] I'm sending you a kiss way more lower than lips...and lower than bosom..."

Then.... again...one month later

"Why did you made me hope for feelings you didn't had !!!! (i'm not adding any "!", it's litteraly that) [...]

Farewell ! never write to me again. 1000 knifes are stabbing me, dont force on it more, goodbye, my love, you were all i had on this planet..."

It's like that for more than 1000 letters.