Was S�betai fat and one-eyed?

Some authors have written that S�betai was fat, but in fact they base that claim on a single passage in the Secret History - where it is said that Chinggis sent after his general in "an iron cart". The writers have then interpreted this as that S�betai was so fat, that he couldn't ride and had to be transported in a strong cart of iron, like some kind of beast!

That hypothesis is completely disconnected and out of context, why then have authors of history books used it? S�betai was evidently not fat - at least he could ride a horse! During the invasion of Kwarizm - according to muslim historians like Juvaini - Chinggis sent for S�betai - who rode several days and nights on his horse to get to his emperor's headquarters as fast as possible. Could such an obese person - who previously is said to have been transported in a cage like an elephant - have done that? No, S�betai couldn't have been fat, or at least not obese

Another claimed attribute of S�betai is that he supposedly was one-eyed. How did he get one-eyed, and where is that stated?

Never heard any of that. It's possible that someone is confusing Subedai with Temujin's old enemy Tarkutai, who was supposedly so fat that he couldn't ride a horse and had to be pulled along in a wagon.

Yes of course, that painting is not contemporary but was made in the 19th century