According to Shia Islam that hadith is not authentic and is fabricated. (explain).
1-Do Shia believe that these three righteous caliphs will end up in Paradise?
No, Shia Muslims does not have such a belief.
Do they believe that this Hadith is not correct?
Yes, Shia Muslims believe that this hadith is not authentic and is fabricated.
If they do believe this, what are the reasons behind this belief?
This hadith contains contradictions :
Imam Ali a.s. considered Killing Talha and Zubair permitted when they didn't returned and Talha and Zubair also considered killing Imam Ali a.s. permitted and they fought each other at the Camel war with Talha and Zubair being killed in this war. They were enemy of each other. How it is possible that two enemy can both go to paradise while fighting and killing each other?
وَمَن يَقْتُلْ مُؤْمِنًا مُّتَعَمِّدًا فَجَزَاؤُهُ جَهَنَّمُ خَالِدًا
فِيهَا وَغَضِبَ اللَّـهُ عَلَيْهِ وَلَعَنَهُ وَأَعَدَّ لَهُ عَذَابًا
But whoever fights [or murders] a believer intentionally - his recompense is Hell,
wherein he will abide eternally, and Allah has become angry with him
and has cursed him and has prepared for him a great punishment.
Talha and Zubair were in the army of Aisha fighting Imam Ali (a.s.) and both were killed in this , then how both side can go to paradise?
Imam Ali a.s. had serious conflicts with Abubakr,Umar and Uthman and each of them had conflicts with each other. Imam Ali a.s. did not accept to follow methods of Abubakr, Umar and Uthman (scanned Sunni ref). Imam Ali a.s. did not participate in the attack of Umar to Iran because had conflict with Umar about the legitimacy of this attack and disagreed this action.
There are many other conflicts between Imam Ali a.s. and Abubakr,Umar and Uthman. Here is a famous sermon from Imam Ali a.s. about them recorded in many Shia and Sunni books:
Sermon 3 Sermon of ash-Shiqshiqiyyah (Including Sunni references)
Also Abdu Alrahman Ibn Auf had serious contradictions with Uthman and was at wrathful with Uthamn until his death.
How so many inconsistency is possible if this hadith is authentic?!
In the camel war before the war starts Imam Ali a.s. called Talha and
Zubair and said them: do you know that the companions of Camel are
Zubair replied: how we are cursed while we are promised to paradise?
Imam Ali a.s.: if I considered you from people of paradise I did not
fight you and did not consider your blood Halal.
Zubair said: have not you heared the hadith of Saiid Ibn Amru Ibn
Nafil from prophet that ten people from Quraish are in paradise?
Imam Ali a.s. said: I heard he said this hadith at time of Caliphet of
Uthman for him.
Zubair said: did he lie?
Imam Ali a.s. said: I do not say anything until you tell their names.
Zubair said the name of nine of them.
Imam Ali a.s. said: who is the 10th?
Zubair said: it is you.
Imam Ali a.s. said: so you admitted that I am from companions of
paradise. but I deny what you claim about yourself and your companions.
Zubair said: do you doubt that Saiid lied?
Imam Ali a.s. said: I do not doubt. I am sure he lied and some of
them are in coffin in the lowest Darak of Hell and at top of that well
is a stone that when Allah wants to punish the people of Hell
removes that stone. I heard this from prophet.
Zubair cried and returned. (ref)
Do they believe that Prophet (SA)'s judgement was wrong?
No, Shia believe prophet was infallible and did not committed any mistake nor any sin nor any Makruh.
References and more study:
The Opinion of Imam Ali (AS) on Caliphate
Caliphate and Imamate
Imam Ali (a.s.) and the Caliphate
Imamate The Vicegerency of the Prophet [s]
About the hadith ten Mubashere hadith
Contradiction inside the hadith of ten Mubashare hadith