Belief in the return of Imam Mahdi (aj) and the incidents that will take place afterwards is considered one of the instances of belief in the unseen that can only be obtained and learned of through authentic hadiths. At the same time, since many hadiths have been subject to dissimulation by the Shia themselves, who feared for their or others’ lives, or have been altered and forged by their enemies, or have just simply been forgotten and wrongly narrated, or have been lost as a result of their not being recorded and taken care of due to lack of resources it isn't possible for the truth-seeking Shia to reach a sure conclusion on the details of the incidents before and after the dhuhur (emergence) of Imam Mahdi (aj) the way that a true Shia would like, especially since there are unjustifiable and unsolvable contradictions between some of the hadiths on this subject. This results in not being able to give a certain and sure answer to your question, especially the first part, and all we can do is make strong conjectures using the hadiths we have at hand. Also, in order to answer your question, it needs to be broken down into separate questions; each question being answered independently.
1- After the rise of Imam Mahdi (aj), how long will it take for his government to be established?
Imam Baqir (as) says: “The twelfth imam (aj) will carry his sword for eight full months, killing the enemies of Allah (swt), until Allah (swt) is pleased.” According to hadiths we have, he will emerge from Mecca and after his chosen ones and companions join him there, they will move towards Iraq and after that Sham (current Syria), then to Beytul-Muqaddas (Jerusalem), and from then on to Europe, Turkey, China, Afghanistan and the rest of the countries, conquering all of them as they go. After eight months of continuous fighting, his government will be established and peace and security will take root all over the world. There is a great difference in hadiths regarding the time span of his rule:
19 years and some months, 7, 40, 10 and 309 years in which each year is equal to 20 or 40 years, and it has been said that after his martyrdom or death (there is a debate whether he will be martyred or will have a natural death), 40 days will pass and then it will become the Day of Judgment and the skies and earth will be gathered so that the hereafter commences.
2- Can poverty still be found after Imam Mahdi’s (aj) government is established?
When the imam (aj) rises and establishes his just government, the earth will give out all of its treasures (plants, minerals etc.) putting them at the disposal of the imam (aj). Everything that is the right of the poor will be returned and divided amongst them justly, to the extent that there will be no poor people left to give charity to. Imam Sadiq (as) says: “As justice is served, the world will thrive; the sky will pour its rain, the trees will give out their fruit and the earth will send out its plants, decorating them for its inhabitants.” Imam Mahdi (as) will order someone to call out to the people asking all of those who are needy to stand. No one will stand other than one person who will stand and say: “I [am in need].” The imam (aj) will tell him: “Go to my treasurer and tell him that the Mahdi (aj) orders you to give me some money.” The treasurer will then say: “Bring your garment.” He will then spread the garment and fill it. When the man throws the full garment over his shoulder, he will regret it and say: “Why should I be the most selfish of the nation of Muhammad (pbuh)?! Why should I not have the humility that everyone else does? He will then return the money to the treasurer, but it will not be received by him, and the imam (aj) will say: “We don’t take back what we have given away.”
The holy prophet of Islam (pbuh) says: “The people will seek individuals in need so that they can give a gift (charity) to them, they separate the alms (zakat) of their wealth, but have no one to give it to, because [at that time], all are made needless by Allah's (swt) grace.”
3- Can any oppressors and tyrants be found at that time?
The agenda of the universal government of Imam Mahdi (aj) will be to establish peace, security and justice everywhere, and to eliminate all traces of corruption, oppression and tyranny; as has been the goal of all prophets; the only difference being that they weren’t able to fully and wholly accomplish those goals, nevertheless they all foretold of a day that would come that such would actually take place. The nature of all human beings, regardless of what time they lived in, has also always been in anticipation of such a day.
Therefore, since Allah (swt), the prophets and the imams are all infallible, and when they make a vow, it is fulfilled and they don’t give their followers false and useless hope and don’t keep them waiting, this promise will also certainly be fulfilled; this promise is so important that it can be considered the goal of creation; the day when all people will unite into one nation under the flag of tawhid, eliminating all traces of kufr, shirk and hypocrisy. The hadiths that directly or indirectly point to this matter are many in number. Imam Askari (as) has been narrated telling his son (aj): “My son! It is as if I see you when Allah's (swt) aid is revealed upon you and the way for relief has been paved and your greatness and magnificence has reached its highest…it is then that the sun of the government of truth will rise and the darkness of falsehood will go away. Allah (swt) will break the backs of tyrants with your hand, revive the sacraments of religion, enlighten the horizons of the world, establish peace and security from coast to coast…your enemies will be made miserable and helpless and your friends will be made glorious and victorious. No oppressors, criminals, rebellious deniers and transgressing enemies and opposers will remain on the face of the earth, because [the Quran says]: “And whoever puts his trust in Allah, He will suffice him. Indeed Allah carries through His command. Certainly Allah has set a measure for everything”.
4- Are there chances of sinning taking place during the reign of the imam?
The main things that factor into one sinning are his urge to accumulate wealth and his need to fulfill worldly desires.
When a person has enough wealth and bears good social status and has a divine mindset and this world and everything in it becomes worthless to him and he reaches certainty regarding the truth of all of Allah's (swt) vows (even though some may not like it and may try to get in the way of some of those vows being fulfilled) and that they will indeed eventually be fulfilled and that the hereafter is for real and there will be judgment and that paradise and everything it encompasses can in no way be compared with this world and its pleasures, there will no longer be any urge and desire to commit any sin, especially since according to some hadiths and the arguments of some great scholars, the life of Shaytan will end with the rise of Imam Mahdi (aj). Also, according to some hadiths, he will be slaughtered by the prophet of Islam (pbuh) in Beytul-Muqaddas (Jerusalem), putting an end to his trickery, temptations and false embellishment [of wrong acts]. At the same time, there will be many things that will prevent the committing of sins, such as the safe and healthy environment the just universal government of the imam will create, and the fact that the people will no longer accept any corruption and sin taking place. This will all result in no one sinning anymore; there are also many hadiths that corroborate this matter as well, namely:
Imam Sadiq (as) describes the time of Imam Mahdi (aj) in such a manner: “Illegitimate relationships, intoxicants and interest (Riba) will all cease to exist, the people will turn to worship and obedience [of the Lord], they will take care of anything they are entrusted with, the bad people will all be annihilated and only the good will remain.” The prophet of Islam (pbuh) says: “Allah (swt) will solve the [Muslim] nation’s problems through the Mahdi (aj), will fill the people’s hearts with worship and obedience and his justice will encompass all. Allah (swt) will destroy lies and lying through him. He will wipe out the spirit of fighting and bestiality and will remove the degrading “collar” of slavery from their necks.”
1- The Holy Quran
2- Ibrahim Amini, Dadgostare Jahan, pg. 388 and 447.
3- Hosein Heydariye Kashani, Hukumate Adl Gostar, pp. 225 and 301.
4- Mohammad Javad Khorasani, Mahdiye Montazar, pp. 188 and 273.
5- Ali Kurani¸ Asre Zuhur, pp. 351 and 370.
6- A group of writers, Adle Montazat, saying 1 and 11.
7- Lotfullah Safi Golpaygani, Montakhabul-Athar.
8- Lotfullah Safi Golpaygani, Emamat va Mahdaviyyat, vol. 3.
9- Kamel Suleiman, Ruzegare Rahayi (translation of Ali Akbar Mahdavipour), vol. 1 and 2.
10- Majlisi, Bihar, vol. 51 and 52.
11- Mohammad Hosein Tabatabai, Tafsir Al-Mizan, vol. 14, pp. 160 and 175.
 See: Lotfollah Safi Golpaygani, Navide Amn va Aman (Emamat va Mahdaviyyat), vol. 3, pp. 28-42.
 Ilzamul-Naseb, pg. 189; Gheybatu Nu’mani, pg. 165; Bisharatul-Islam, pg. 199, quoted by Ruzegare Rahayi, vol. 1, pg. 478, hadith 585.
 See: Ruzegare Rahayi, vol. 2, pg. 470 and 536; Biharul-Anwar, vol. 52, pg.309 and 392.
 Ruzegare Rahayi, vol. 2, pg. 601 and 658; Bihar, vol. 2, pg. 279 and 392.
 Bisharatul-Islam, pg. 71, quoted by Ruzegare Rahayi, pg. 639, hadith 882.
 Muntakhabul-Athar, pg. 147; Musnad Ahmad, vol. 3, pg. 37; Al-Sawa’iqul-Muhriqah, pg. 164; Yanabi’ul-Mawaddah, vol. 3, pg. 135, quoted by Ruzegare Rahayi, pg. 642, hadith 894; also, same book, pg. 596 and 598.
 Ilzamul-Naseb, pg. 230; Bisharatul-Islam, 256; Ruzegare Rahayi, pg. 598, hadith 780.
 Hud:118 and 123.
 Biharul-Anwar, vol. 5, pg. 35; Wafatul-Askari, pg.49, quoted by Ruzegare Rahayi, vol. 1, pg. 530-531; also, same book, pg. 73, hadith 5, pg. 625, hadith 844.
 See: Al-Mizan, vol. 14, under verse 36 of surah Hijr, pp. 160, 161 and 175.
 Muntakhabul-Athar, pg. 474; Ilzamul-Naseb, pg. 228; Al-Malahimu wal-Fitan, pg. 54, quoted by Ruzegare Rahayi, vol. 2, pg. 600.
 Bihar, vol. 51, pg. 75; Al-Malahimu wal-Fitan, pg. 56; Gheybah of Sheikh Tusi, pg. 114, and…