In Heart of Darkness, why does Kurtz go to the Congo?

Kurtz is initially sent to the Congo to bring back ivory for the Company. This was standard procedure in that era; Kurtz was a "boss" overseeing white men and native slaves, and his output of ivory dictated how much support (difficult and expensive to send) he received in turn from the Company. However, Kurtz discovers that he values the power he holds over his men more than his rewards for service, and is overpowered by that lust.

But his soul was mad. Being alone in the wilderness, it had looked within itself, and, by heavens! I tell you, it had gone mad. I had -- for my sins, I suppose -- to go through the ordeal of looking into it myself. No eloquence could have been so withering to one's belief in mankind as his final burst of sincerity.
(Conrad, Heart of Darkness, gutenberg.org)

Kurtz is obsessed with his own power. He can control men through his voice and his physical prowess, and has set himself up as almost a god among his men and the natives. Only his sickness causes him to lose that power. In the end, he did not care about the Company or even truly about his Intended, but instead cared only for the feeling that wielding that power created. While he may have recovered his sanity in time, his ill-health prevented that, and in the end Kurtz's journey was destroyed partially by his own obsession. 

Why does the manager, viewing Kurtz as a threat, accompany Marlow to find him?

The manager is one of Marlow's assigned "devils" in the company.  As he overhears the manager speaking with his brother, he realizes that the manager in now way has the well-being of Kurtz in mind. He is afraid for his own job and is jealous of his successes.  Unfortunately his rogue status has given him a reason to interfere with the workings of the Inner Station.

The manager knows that Kurtz is ill, close to death, and he has been depriving him of shipments of necessary medication.  At this point, he needs Marlow to do his dirty work:  to find Kurtz and to get him out of the way.  The manager is banking on the fact that once Kurtz is located, he can be removed from his post for "medical" reasons and not cause the company any embarrassment.  In fact, his dream comes true when Kurtz dies en route.

The manager needs someone to find and remove Kurtz from his station before he reflects poorly on the direction and operation of his (the manager's) post.

In Heart of Darkness, why does the manager choose Marlow to find Kurtz?

In Joseph Conrad's Heart of Darkness, Captain Marlow is assigned a mission to find Kurtz, an ivory hunter who has made a lot of money for the Company. Marlow is chosen partly because of his competence and partly because he is an outsider.

The manager of the first station, who is never named, strikes Marlow as odd, and Marlow later concludes that the man is empty inside -- he has no heart or soul, and although he is intelligent and able in his job, he has no concern for anything but the bottom line:

“He began to speak... I had been very long on the road. He could not wait. Had to start without me. The up-river stations had to be relieved. There had been so many delays already..."
(Conrad, Heart of Darkness, eNotes eText)

The manager is entirely concerned with the business of ivory. Kurtz, who is a massively prolific ivory hunter, has been out of contact for a long time, and so the manager accompanies Marlow to find him. However, because of Kurtz's reputation for strange and unethical behavior, the manager is also planning to scapegoat him for troubles on the river:

"'We have done all we could for him—haven't we? But there is no disguising the fact, Mr. Kurtz has done more harm than good to the Company. He did not see the time was not ripe for vigorous action.'"
(Conrad, Heart of Darkness, eNotes eText)

Marlow, who is a very pragmatic man, is able to complete the mission, but many conversations and plans go on in secret, some of which he overhears. His status as an outsider to the Company allows him objectivity, but it also give the Company an out if the mission goes badly; they can blame it on him instead of their own men.

In Heart of Darkness, what was Kurtz's original motive for entering the jungle?

Because of the high profitability and difficult work of hunting ivory in the jungle, men who could command respect and power were well-suited to lead hunting teams. However, Kurtz's original intent was not simply to be a successful ivory hunter, but to win the heart of a woman, whose family rejected him because he was poor.

I had heard that her engagement with Kurtz had been disapproved by her people... He had given me some reason to infer that it was his impatience of comparative poverty that drove him out there.
(Conrad, Heart of Darkness, eNotes eText)

Determined to prove them wrong, he entered the jungle and became very successful, bringing in "as much ivory as all the others put together." However, the power he commands alters his mind and eliminates his conscience, and although he continues to send ivory back, he also sets himself up as a god to the natives. It is his failure to realize his original goal that leaves him with nothing to strive for except his own personal power and glory. Had he remained sane and become wealthy, he might have been able to leave the jungle and return to civilization.