Short description of the plot of "Five weeks in a balloon" (second and third chapters)

On tThe next day an article appeared in the “Daily Telegraph” and which revealed all the plans of Samuel Fergusson. ("...all Samuel Fergusson's plans" would be better).

The plan was to cross Africa from east to west on in a balloon.

This provoked a great sensation., mMany doubted in Samuel Ferguson and in his intentions, many doubted even in the possibility of crossing the African continent oin a balloon.

Samuel was mocked by some newspapers, but also he was praised by others.

You don't need "also" as that is understood by "others".
You could also say "... by some newspapers; however, others praised him."

There wereas a lot of bets on the result of his campaign: could he to start his travel or notset off successfully on his trip, could he cross the continent or not, if not – in which part would he will fail, would he siurvive, etc.

You could also say: "There was a great deal of betting on the outcome of his campaign..."

But in spite of all this, he continued his preparations.

Many was were encouraged by his resolution and courageous, and offered themselves to make him company to accompany him and share with him the joy of victory or the bitterness of failure, but her refused them all without explaining the reason.

"... without explaining why" or "without giving the reason" would be a little better.

Many inventors proposed to him some of their mechanisms and gears but he turned them all down without explaining if he has whether he had his own inventions.

This friend was called Dick Kennedy and he was a scotch Scottish hunter.This friend, a Scotsman called Dick Kennedy, was a hunter.

Re "scotch" -- people are always referred to as Scottish or Scots, NEVER Scotch.
Only things may be described as "Scotch", e.g. a Scotch egg.
"Scotch" on its own is taken to mean whisky distilled in Scotland.

Dick and Samuel had met on their military service in India where they had been in one the same regiment.

You don't need to put "their" in. It is understood from the context.

After military service they had continued to see each other and Samuel often had often visited his friend in at his home near Edinburgh, when he was not in away on another expedition.

But in the last two years Samuel had neither participated in any expedition nor expressed a desire to participate in one. sSo Dick had made a come to the conclusion that Samuel had satisfied his thirst for adventures and from now would live a quiet life. , and Dick was glad because of this, as he had thought the opinion that a man couldn’t travel all his life among savages and cannibals and where any misfortune could happen in at any moment.

It is too long a sentence. It is better as a few shorter sentences.
"he had thought" -- this is too brief, really. It is as if he thought it once and then forgot it. This is more of a long-standing opinion.

But this morning he saw in the “Daily Telegraph” Samuel’s plan to cross Africa oin a ballon and this enraged him., and hHe got formed an idea to prevent his friend from undertaking this mad adventure.

You could say instead "he formed an idea to DISCOURAGE his friend from this mad adventure", but if you use "prevent", you prevent someone from DOING something.

He took the train to London and the next day he was before outside the door of his friend’s house.

"... the next day he STOOD outside the door of his friend's house" would be better style than "was".

Samuel was not very surprised to see his friend before in front of him but he was a little bit somewhat staggered by his friend’s reaction on to his project.

"... to see his friend confronting him..." is also possible, because Mr Kennedy was angry with him.
You cannot be "a little bit" staggered. "Staggered" is quite a strong word, not one for half measures. But I think "somewhat staggered" is all right.

At first hei thought that Dick was angry because he had not been invited him to participate in the preparations of for the adventure. He therefore and so he hastened to assure him Dick that whe would have surely sentd him a letter without fail after the preparations were done would have finished as he intended to take him Dick on the a balloon as a crew member.

I put Dick's name in a few times just to make it clear -- there are a lot of instances of "he" and "him" in this passage!
"On" the balloon is correct in this last sentence.

Dick became even more terrified with at this statement and he said that he would never have never accepted an invitation to participate in such an insane expedition.

or "... he would never have agreed to participate ..."

Samuel demanded Dick's attention. to hear him attentively, hHe explained to Dick that the travelling by air is the safest way to travel across Africa as many previous expeditions had failed due to an land accidents on a surface as the main danger had come from the savages, who killed many explorers.

But in fact he was not quite assured in of the safety of this expedition, and he still had no intention of to participating e in it, and moreover, even more he still planned to prevent his friend from undertaking getting into it.