Are the historical books of Herodot or Polybius or any other historian from ancient history usually correct? Did they add some 'drama' in their books or did they desire to gain favor of those of high ranks, by praising them, by spreading their fame in some of their texts? Did we ever have proven historical inaccuracies in one of their books? For example, Polybius has written that Scipio Aemilianus cried at the destruction of Carthage. How can this be true? How did he know that as no one saw him?
Historians engage with such sources all the time. Just like any other historical source, they are evaluated in relation to other available evidence. It is true that ancient historians had different standards of evidence then modern ones, and this must be taken in to account. But all sources can include distortions and that is in no way unique to these sources.
If you want to know specifically whether Scipio Aemilianus really cried at the destruction of Carthage, you are going to look at other sources besides Polybius to help you determine that. Can you find out what source(s) Polybius took that information from? Can you least find independent sources (ones that could not be based on Polybius) that claim the same thing? Are there secondary sources (modern historians) who support this claim? This kind of thinking is a big part of what historians do.
Question: Are the historical sources from the ancient history trustable?
Controversial topic. There is no blanket finding on historical texts accuracy or inaccuracy which enjoy's consensus support among historians. Historians generally support the critical analysis of historical texts matching them against broad sources and references to determine each documents veracity.
If we examine Herodotus. Plutarch wrote "The Malice Of Herodotus", an essay criticizing The Histories for their inaccuracies and misrepresentations.
Herodotus is honored as the father of History. He traveled to some of the lands he wrote about and interviewed locals. He also conducted analysis of his history to attempt to maintain it's correctness across conflicting sources. Similar to what modern Historians do without access to all of our resources. Still Herodotus is Not to be taken as objective fact. Today we know more than Herodotus did. We have the advantage of archaeological finds, other writings, and scientific analysis which Herodotus did not.
Examples of Herodotus getting his facts wrong:
Book One of The Histories, Herodotus says Babylon had 100 bronze gates around the city, that Babylon’s walls were 100 meters high, 22 kilometers (14 mi) long, and 50 meters (164 ft) thick. He also says he saw a deep, water-filled trench that surrounded the entire perimeter of the city. Archaeologists do not support Herodotus's "observations". Babylon only had 8 gates and it's walls were not nearly as formatable or vast.
In Book Two, Herodotus writes about Egyption embalming techniques. With modern medical technology we know Herodotus attributed embalming techniques were wrong. "A lot of his(Herodotus's) accounts sound more like tourist stories, so we're reticent to take everything he said at face value," co-author of the study Andrew Wade told LiveScience.
In Book Three (Book 3, passages 102 to 105), Herodotus claimed giant insects(ants), the size of "foxes" in Persia were used in the mining of gold. Not really something which stands up to the test of credulity. Although some suggest Herodotus was describing squirrels or Himalayan marmot. Even so, That's not an ant.
In Books Three and Four, Herodotus makes multiple mentions of a race of Cyclopes and griffons indigenous to Europe. We've found no evidence of such a race of single eyed giants, nor of the griffons who Herodotus said guarded large hoards of gold.
In his An Account of Egypt Herodotus speaks a lot about the flooding of the Nile river. He evaluates and dismisses three theories on why the river floods seasonally before proposing his own. In his theory, Herodotus gets the months wrong. Herodotus says the Nile retreats in the summer months due to the heat of the sun evaporating the water. Only the Nile actually floods in the summer. Herodotus Speculates about Egypt
All throughout his histories Herodotus, who was Greek, is overtly biased in favor of the Greeks. Specifically in his description of the Battle of Marathon, Herodotus says only 192 Athenians were killed vs 6,400 Persians. Modern estimates place Athen's casualties at 1000-3000, (Battle of Marathon). Most historians chalk Herodotus's numbers up to exaggeration. Also Herodotus claimed that the 192 Greek warriors who fell at Marathon were buried in the middle of the battlefield, on the planes of Marathon. Such a burial site, supported by carbon dating doesn't exist. The long time candidate for this site at Marathon (soros) actually predates Marathon based on the carbon dating of ceramics recovered from mound. Modern historians today don't even agree the actual battle occured on the plains of Marathon, but rather debate a different site miles removed from the soros mound.
Lion cubs don't actually claw their way out of their mother's wombs, as Herodotus describes and the number of offspring of a species are not related to how fierce or dosil their species is as Herodotus reports.
This is a great question. Unfortunately like many great questions there is no consensus which broadly categorizes ancient texts. While some ancient historians like Herodotus had some understanding of objective history, even if they did not always live up to them; many historians did not. Each document needs to be evaluated on it's own merits and critically analysed to discover it's objectivity and factual merits.