All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad  is His Slave and Messenger.
  is His Slave and Messenger.
The first Hadeeth was narrated by Abu Al-Layth As-Samarqandi  in his book "Bustaan" on the authority of Abu Umaamah Al-Baahili
  in his book "Bustaan" on the authority of Abu Umaamah Al-Baahili  that the Prophet
  that the Prophet  said: "Allaah detests the fat scholar."
  said: "Allaah detests the fat scholar." 
Shaykh Mashhoor Hasan cited all the chains of narration of this Hadeeth and declared it weak in his revision of Al-Muwaafaqaat. 
Shaykh Ash-Shinqeeti  also declared it weak in his book Adhwaa’ Al-Bayaan. He commented on it saying, "What the Hadeeth mentions of dispraising overeating and over drinking and acquired obesity (as opposed to natural or genetic obesity) is obvious and has many supportive evidences, such as the Hadeeth: 'It is sufficient for a believer to eat a few mouthfuls to keep his spine straight.'"
  also declared it weak in his book Adhwaa’ Al-Bayaan. He commented on it saying, "What the Hadeeth mentions of dispraising overeating and over drinking and acquired obesity (as opposed to natural or genetic obesity) is obvious and has many supportive evidences, such as the Hadeeth: 'It is sufficient for a believer to eat a few mouthfuls to keep his spine straight.'" 
An-Nawawi  wrote, "The blameworthy obesity is that which one acquires. This does not apply to a person who is naturally overweight (genetic obesity) ..." [Sharh Muslim]
  wrote, "The blameworthy obesity is that which one acquires. This does not apply to a person who is naturally overweight (genetic obesity) ..." [Sharh Muslim] 
Hence, the blameworthy obesity is that which is acquired due to overeating and gluttony and not congenital overweight. 
The fact that a scholar is obese and eats much does not prevent learning at his hands as long as he is trustworthy and his knowledge is reliable. Ibn Rajab Al-Hanbali  wrote in his book Sharh Hadeeth Labbayka Allaahumma Labbayka, "Abu Al-Husayn ibn Bashshaar was once asked, 'Can the friend of Allaah be obese?' He replied, 'Yes, if he is trustworthy.'"
  wrote in his book Sharh Hadeeth Labbayka Allaahumma Labbayka, "Abu Al-Husayn ibn Bashshaar was once asked, 'Can the friend of Allaah be obese?' He replied, 'Yes, if he is trustworthy.'" 
As for the second Hadeeth, it was declared weak by Al-Albaani  in his book Silsilat Al-Ahaadeeth Adh-Dhaʻeefah. The Hadeeth reads,
  in his book Silsilat Al-Ahaadeeth Adh-Dhaʻeefah. The Hadeeth reads, 
"It has been narrated on the authority of Anas ibn Maalik  that a boy was martyred in the Battle of Uhud, and it was found that he had a stone tied to his stomach because of hunger. His mother wiped out the dust from his face and said, 'Glad tidings of Paradise to you, my son.' The Messenger of Allaah
  that a boy was martyred in the Battle of Uhud, and it was found that he had a stone tied to his stomach because of hunger. His mother wiped out the dust from his face and said, 'Glad tidings of Paradise to you, my son.' The Messenger of Allaah  said "How do you know! Perhaps he spoke of that which did not concern him, or he was miserly with something that he could do without harming himself."
  said "How do you know! Perhaps he spoke of that which did not concern him, or he was miserly with something that he could do without harming himself." 
Al-Albaani  declared a similar Hadeeth authentic and cited it in his book As-Silsilah As-Saheehah. It reads,
  declared a similar Hadeeth authentic and cited it in his book As-Silsilah As-Saheehah. It reads, 
"It has been narrated on the authority of Kaʻb ibn ʻUjrah  that the Prophet
  that the Prophet  missed Kaʻb and so asked about him; they replied, 'He is ill.' So the Prophet
  missed Kaʻb and so asked about him; they replied, 'He is ill.' So the Prophet  left and walked until he came to him. When he entered upon Kaʻb, he said, 'Receive the glad tidings, O Kaʻb!' So his mother said, 'O Kaʻb, congratulations! For you is Paradise!' The Prophet
  left and walked until he came to him. When he entered upon Kaʻb, he said, 'Receive the glad tidings, O Kaʻb!' So his mother said, 'O Kaʻb, congratulations! For you is Paradise!' The Prophet  said, 'Who is the one who is taking an oath upon Allaah [that Kaʻb has a place in Paradise]?' Kaʻb replied, 'She is my mother, O Messenger of Allaah.' So, he
  said, 'Who is the one who is taking an oath upon Allaah [that Kaʻb has a place in Paradise]?' Kaʻb replied, 'She is my mother, O Messenger of Allaah.' So, he  replied, 'And how do you know, O mother of Kaʻb [that he is going to Paradise]? Maybe Kaʻb spoke about something which did not concern him or he withheld giving something which was insignificant?'"
  replied, 'And how do you know, O mother of Kaʻb [that he is going to Paradise]? Maybe Kaʻb spoke about something which did not concern him or he withheld giving something which was insignificant?'" 
Al-Ghazaali  wrote,
  wrote, 
"It means that Paradise is being prepared for those who would be admitted into it with no reckoning; however, he who had engaged in talking about that which did not concern him shall be held accountable for his words. If his sayings involved any prohibited acts, Paradise shall never be prepared for someone who shall be examined thoroughly at the Reckoning for it is an aspect of punishment..." [Al-'Ihyaa'] 
Al-Mullaa Al-Qaar'i  wrote,
  wrote, 
"Withholding something which is insignificant pertains to what he should give in the money-related acts of worship or issues of knowledge or worldly transactions..." [Mirqaat Al-Mafaateeh Sharh Mishkaat Al-Masaabeeh] 
Allaah Knows best.