For me, the main problem is to believe that with technology of 1969, they could go to moon and gothen back home.

The comma after 1969 was added because that is where two dependent clauses connect. The "then" was added because they would go back home after the going to the moon, rather than going home just in general. The "then" helps specify that they went to the moon first, travelling home second. The "go" was removed because with the first "go", we already know that they are travelling somewhere, so the second one is repetitive!

Now it seems incredible that thisa rocket with this technology could to take off.

The reason the "it" was added was to specify what seemed incredible. "seem" -> "seems" because "seem" is not a stand alone word. Some correct usages would be "does seem", "can seem", or "to seem". It is usually found with another word before it that puts it in context.

"this" should be "a" because "this" refers to a specific one, but your language of "with this technology" implies that there could be similar rockets with the same technology that would seem just as questionable upon take-off.

"to" is unnecessary here since it is a preposition, and "to take off" is not a preposition in the right context. For example, "to" could be used like any of the following:
"to the windows"
"to the walls"
"to the house"
Prepositions such as "to" or "down" are used in a phrase that specifies a place as the last word, as you saw in the examples provided. Here's some for "down":
"down the street"
"down to the market"
In conclusion, you can just end with "could take off" .

:)

There is a anecdote about an astronaut, who I think was Neil Armstrong, about some datas from computer that he not verydid not trust.

Instead or in addition to the first comma, you would put "who".
Data in itself is plural, so no need to end with an 's'!
"did" was added in because you needed a verb. "not" in itself is a bit of an extra to the verb "did" in this scenario.
"did" as a stand alone means that someone performed something, so adding the negative "not" on to the end means that they did not perform something.
The "something" in your sentence would be Neil Armstrong's trust! So "did not trust" would be correct :)

He asked for a checking of the datas to one of the women that did the manual calculation.

If I believe the man reached the moon, it is because the URSS did not deny the facts.

"it" needed to be added because you needed to specify: what is because the URSS did not deny the facts? "it" being your belief in the facts.

Sometimes, I think theat man really reached the moon, but sometimes, I have serious doubts.

For me, the main problem is to belive that with technology ofrom 1969, they could go to the moon and go back home.

You can add "somehow" between could and go to express your doubt.

The computer of Apollo 11 worked ato 2 MHz speed, with 2 Kkb deof RAM and 32 Kkb of ROM.

Now, it seems incredible that this rocket with this technology could to take off.

There is an anecdote about an astronaut, I think waspossibly Neil Armstrong, about some datas from the computer that he could not very trust.

He asked a checking of datas toverification of the data from one of the women that did the manual calculation.

Checking isn't used here usually. Other words like verification or inspection could work.

If I believe theat man reached the moon, it is because the URSS did not deny the facts.