Yes, it does.
To know why we have to reborn even if we pay for our sins in the same life we need to understand the three types of Karma, viz. sanchita or arjita(stored/preserved), prārabdha(preordained) and kriyamāna or āgama (accumulating):
prārabdhāgāmyarjitāni karmatrayamitīritam [Varah Up. - 12]
Prārabdha, āgami and arjita, these are the three karmas.
The coming of existence (creation) and ceasing of existence (dissolution) have been going on infinitely. Hence, a jivatma also takes birth and dies infinite times in it. As a jivatma continues his life, he engages in numerous kinds of actions both good and bad. But his single life time may not be enough to experience the results of all his actions. And because a reaction must follow an action, to experience the reaction of his own actions the jiva has to take birth again accordingly.
Now the thing is, because a jivatama takes countless births, his actions are also countless. So all the unexperienced pending reactions of the jivatama that are due to be experienced are stored as Sanchita Karma. So sanchita karma of a jiva is too large, nearly infinite.
Only a portion of the sanchita karma can be experienced in one life time. So that portion of karma which determines most events and type of life of a jivatma is known as Prarabdha Karma. It is this prarabdha karma that determines the species of life and major events in life.
Because a jivatma is not only experiencing reactions to his past life actions (like sudden accident, winning of lottery, etc.) but also engaged in newer ones, all his present life actions get accumulated and known as kriyamana karma. These karmas will either yield result in the same life or if they do not get experienced then they will get stored up as sanchita karma to influence the future lives of the jivatma.
Now that we know these three types of Karma, the answers to your questions can be understood easily:
If we pay for our sins in the same life, then why do we reincarnate?
It is because, even if we pay for some of our deeds, we have numerous other reactions of psat life still in pending to be experienced. So even if our karmic account of this life stays clean, to balance out our past actions we have to take birth again unless we clear them out some way in the present life.
If karma of past life is not considered then why people have troubles in their lives?
This question indeed suggests the fact that Karma of past life do affect the present life. For example, sometimes even good persons face accidents, misfortune etc. and some bad people, even though engaged in bad actions, move freely. This happens all because of prarabdha karma. If a person has in his prarabdha to face accident, then doesn't matter how good he is in his present life, he will certainly face it to balance out his past life action. Similarly, if a person has in prarabdha to enjoy a luxurious life, then he will certainly do so even if he does bad actions.
Karma goes very deep (gahanā karmaṇo gatiḥ- BG 4.17). Hence, understanding all its implications becomes difficult and complex. But I hope this answers the two questions you have asked.