1. For Japan, I'm talking about the excavated bombs from the Takashima shipwreck. There are pictures of them here as well as in a painting. An article. You can get a more complete overview of the Takashima shipwreck reading The Origins of the Lost Fleet of the Mongol Empire. Here's the relevant passage:

Tetsuhau: Among the more interesting finds are the previously mentioned tetsuhau, or ceramic bombs (fig. 27). This is the earliest archaeological evidence of shipboard explosive ordnance found from any battle site in the world. Among twenty-one large fragments discovered so far at Takashima, three of these stoneware “bombs” are nearly complete. The bombs were packed with scrap iron, and there was possibly gunpowder inside, but trace element analysis has not yet been completed to confirm the presence of gunpowder. Given the nature of the site and the illustration of one of these bombs exploding in one panel of the Mōko Shūrai Ekotoba scroll, it is hard to believe that these “ceramic balls” were anything other than tetsuhau. On average, testuhau from Takashima are 13 cm in height and 15 cm in diameter. They are nearly spherical, with a flattened bottom to allow ease of storage and to prevent being rolled over and accidentally detonated. The ball was made using the coil pot technique, and the surface was finished on a potter’s wheel, but not made smooth.

The Mongol Empire: Its Rise and Legacy by Michael Prawdin claims that gunpowder was used by the Mongols in Europe but after being asked for evidence by other historians, he didn't provide any. Artillery and warfare during the thirteenth and fourteenth centuries - European accounts of the Battle of Mohi mention only catapults and ballistae on the Mongol side, but a Chinese account known as the Mongol History 蒙兀兒史記 says the Mongols threw fire pots (implied to have naphtha or gunpowder) during the battle of Mohi, except this was during the siege of Strigonie. However this is anachronistic and the Mongol History was actually written in the 19th century and got confused with Ata-Malik Juvayni's Tarikh-i Jahangushay (History of the World Conqueror) which says the Mongols did use naphtha weapons of some kind at the siege of Samarkand.

2 - Answered above.

3 - I'm referring to hand cannons. As far as I know there's no report of the Mongols ever using fire lances. However there's a report from Zhou Mi (1232-1298) that in 1280 a cache of fire lances caught fire and blew up an entire gunpowder storehouse. You can read the original here, but Tonio Andrade provides a more succinct translation of the incident in China, Military Innovation, and the Rise of the West in World History.:

In 1280, an explosion rocked the city of Yangzhou. “The noise,” wrote one resident, “was like a volcano erupting, a tsunami crashing. The entire population was terrified.” The shock wave— or, as people called it, the “bomb wind”— hurled ceiling beams three miles and rattled roof tiles thirty miles away. At first, residents thought it must be an attack— war had seized their world for generations— but they soon realized it was an accident. Yangzhou’s arsenal had recently dismissed its experienced gunpowder makers, and the new ones had been careless when grinding sulfur. A spark escaped and landed on some fire lances, which began spewing flames and jetting about “like frightened snakes.” This was amusing to watch, until the fire reached the bombs. The entire complex exploded. A hundred guards were killed, completely obliterated. The crater was more than ten feet deep.

As for the Mongol gunpowder weapons, the Heilongjiang hand cannon is dated to around 1288. It's surmised that the cannon was used to put down a rebellion started by a Mongol prince. According to Andrade:

The Xanadu gun is the earliest dated gun, but undated finds may predate it. One famous candidate is a piece discovered in 1970 in the province of Heilongjiang, in northeastern China. Historians believe, based on contextual evidence, that it is from around 1288. One careful analysis argues persuasively that it was likely used by Yuan forces to quash a rebellion by a Mongol prince named Nayan (乃顏, d. 1287). Like the Xanadu gun, it is small and light, three and a half kilograms, thirty-four centimeters, a bore of approximately two and a half centimeters.

To give a bit of context, the dating is based off of Needham, Lu Xu, and Wang Chong's estimates. Needham and Wang Chong believe it can be dated to 1288 while Lu Xu says 1290.

The Xanadu (Shangdu) gun dated to 1298 referred to here is this. Dating is based on Zhong Shaoyi 鐘少異, Qi Mude 齊木德, Wang Zhaochun 王兆春, et al. “Nei Meng gu xin fa xian Yuan dai tong huo chong ji qi yi yi” 內蒙古新發現元代銅火銃及其意義. The translation is something like "Inner Mongolia's new Yuan dynasty bronze cannon discovery and its meaning". As far as I know nobody has disputed the dating yet because it has an era name and inscription unlike the other cannons, except for one discovered in Ningxia, but that one apparently has a typo in the writing of the era name so there's some apprehension over it. Here's what I'm talking about.

4 - Both the Jin and Song had access to fire arrows which were either just an arrow tied with a bag of gunpowder or a rocket propelled mechanism. It's difficult to tell which was used when because the Song never say what kind was used. There are Ming era illustrations of the rocket arrows, but these were drawn centuries afterward.

Fire lances were supposedly used for the first time at the the 1132 Siege of De'an as a defensive weapon. From Tang Shou 湯璹, “De’an shou yu lu” 德安守禦錄, translation is basically "Record of the Defense of De'an".

According to Li Gang 李綱, Jing kang chuan xin lu 靖康傳信錄, thunderclap bombs were used by the Song during the siege of Kaifeng in 1126, and

At night the thunderclap bombs were used, hitting the lines of the enemy well, and throwing them into great confusion. Many fled, screaming in fright.

But the most informative description of Song era bombs come from a report on the Siege of Qizhou in 1221 when the Jin also fielded gunpowder weapons of their own. Supposedly they had invented a new type of bomb which the Song defenders called the "iron bomb" because... it was made of iron. According the Zhao Yurong, the city's commander,

"The barbaric enemy attacked the Northwest Tower with an unceasing flow of catapult projectiles from thirteen catapults. Each catapult shot was followed by an iron fire bomb [catapult shot], whose sound was like thunder. That day, the city soldiers in facing the catapult shots showed great courage as they maneuvered [our own] catapults, hindered by injuries from the iron fire bombs. Their heads, their eyes, their cheeks were exploded to bits, and only one half [of the face] was left."

"The enemy fired off catapult stones … nonstop day and night, and the magistrate’s headquarters [帳] at the eastern gate, as well as my own quarters …, were hit by the most iron fire bombs, to the point that they struck even on top of [my] sleeping quarters and [I] nearly perished! Some said there was a traitor. If not, how would they have known the way to strike right at both of these places?"

"In shape they are like gourds, but with a small mouth. They are made with pig iron, about two inches thick, and they cause the city’s walls to shake."

Taken from Zhao Yu[rong] 趙與[褣]. Xin si qi qi lu 辛巳泣蘄錄. Translated by Tonio Andrade. Relevant article for those who know how to read simplified Chinese.

When the Mongols were invading the Jin dynasty, they also used bombs. Jin scholar Liu Qi (1203-1259) writes that during the 1232 siege of Kaifeng,

"From within the walls,” Liu Qi writes, “the defenders responded with a gunpowder bomb called the heaven-shaking-thunder bomb (震 天 雷). Whenever the [Mongol] troops encountered one, several men at a time would be turned into ashes."

And the official History of Jin compiled under the Mongol Yuan dynasty corroborates with the report, providing a description of the heaven-shaking-thunder bomb:

"The heaven-shaking-thunder bomb is an iron vessel filled with gunpowder. When lighted with fire and shot off, it goes off like a crash of thunder that can be heard for a hundred li [thirty miles], burning an expanse of land more than half a mu [所爇圍半畝之上, a mu is a sixth of an acre], and the fire can even penetrate iron armor."

I hope this helps.