Addressing your questions one by one.
Q. Where was Ashwatthama when Pandavas spread the false news of his death? On that day, was Ashwatthama too far from his father like Arjun was when Abhimanyu died?
Yes, Ashwatthaman was far away from Drona and not at his sight. This is what gave his chance to Pandavas to plan the way to slay Drona. Ashwatthaman didn't had any idea what was happening to his father. He was in the middle of the battle. That's the reason he asked why Kaurava forces are retreating.
Only Drona's son, Aswatthaman, like a huge alligator coming up against the current of a stream, rushed against his foes. A fierce battle took place between him and many warriors headed by Sikhandin and the Prabhadrakas, the Panchalas, the Chedis, and the Kaikeyas. [Section CXCIV, Drona Parva].
That why he asked Duryodhana why Kaurava forces were retreating instead of fighting.
Proceeding towards Duryodhana, Drona's son, approaching the Kuru king, said, 'Why, O Bharata, are the troops flying away as if in fear? Although flying away, thus, O monarch, why dost thou not yet rally them in battle? [Section CXCIV Drona Parva]
Q. If so then, how Dronacharya could believe that Bhima killed him and came back from that site?
Bhimasena killed an elephant named Ashwatthama from his own army and shouted that Ashwatthama is dead. Dronacharya didn't believe his words. He knows the prowess of his son and continued to fight.
Then the mighty-armed Bhima, O king, slew with a mace a foe-crushing, terrible and huge elephant named Aswatthaman, of his own army, belonging to Indravarman, the chief of the Malavas. Approaching Drona then in that battle with some bashfulness Bhimasena began to exclaim aloud, 'Aswatthaman hath been slain.' That elephant named Aswatthaman having been thus slain, Bhima spoke of Aswatthaman's slaughter. Keeping the true fact within his mind, he said what was untrue, Hearing those highly disagreeable words of Bhima and reflecting upon them, Drona's limbs seemed to dissolve like sands in water. Recollecting however, the prowess of his son, he soon came to regard that intelligence as false. Hearing, therefore, of his slaughter, Drona did not become unmanned. Indeed, soon recovering his senses, he became comforted, remembering that his son was incapable of being resisted by foes. [Section CXCI, Drona Parva].
After this incident, fought with more rage. Then sages appeared at that place in their subtle forms and said that he was fighting unrighteously. They informed that time has come to end his life. Hearing their words, he has realized that his last hour has come. He became extremely cheerless and inquired Yudhishtira whether his son was slain or not. Dronacharya firmly believed that Yudhishtira would never speak a lie. That was the reason he asked Yudhishtira only and ignored Bhimas's words. After making up his mind, Yudhishtira uttered loudly that Ashwatthama is dead adding indistinctly the world elephant (after the name).
Hearing these words of their as also those spoken by Bhimasena, and beholding Dhrishtadyumna before him, Drona became exceedingly cheerless in battle. Burning with grief and exceedingly afflicted, he enquired of Kunti's son Yudhishthira as to whether his son (Aswatthaman) had been slain or not. Drona firmly believed that Yudhishthira would never speak an untruth even for the sake of the sovereignty of the three worlds. For this reason, that bull among Brahmanas asked Yudhishthira and not any body else. He had hoped for truth from Yudhishthira from the latter's infancy. [Section CXCI, Drona Parva].
Q.By seeing that Drona gave up his weapons and Dṛṣṭadyumna is approaching him, why nobody from Kaurvas Army revealed the truth to their Army chief?
After Bhimasena said his son is no more, he didn't believe and continued his fight more vigourously. So, no one from Kaurava troops might not have revealed the truth.
It doesn't take much time to utter a sentence that someone is dead. Drona felt distressed and renounced his weapons as soon as he heard Yudhishtira. It is already explained above that he had firm faith in Yudhishtira that he never utters a lie. After getting that confirmation from Dharma Raja, he didn't ask anyone else. Even someone said his son is alive, he would not believe. He gave up weapons when Yudhishtira said Ashwatthama is dead. He also realised that his last hour on earth has arrived by the words of Sages. So, he never asked others confirmation or checked for his son himself.
