Mongol bows and specific tactics are definitely tough to get a clear picture for, especially since many Mongol battles started with the enemy at a heavy strategic disadvantage, making it difficult to examine 'even fights'. It seems very likely that Mongol tactics and equipment were comparable and based on Khitan/Jurchen tactics and equipment, though perhaps further evolved given their consistent success against the Jurchen. We know from Mongol sources that they had a number of different tactical formations they could switch between during combat, which was controlled by the commander using flags and signals.
Its very dangerous to use western or Islamic examples (i,e Carrhae or Manzikert) to relate to Mongol battles, because Mongol and Manchurian-style battles seemed to have developed very differently than the more skirmishing style that is represented in western works, such as the Frankish knights and infantry in the first crusade being impervious to Turkish arrows. Specifically, the Mongols fought very aggressively, and the battles were often over very quickly. The Rus army at the Kalka in 1223 was routed almost upon contact [all but the front ranks would have been in marching formation], with the Mongols immediately delivering a lance charge. The Jin army at Yehuling was in 1211 routed by noon with a lance charge, Liegnitz (though terribly unreliable) does not appear to have been a particularly drawn out battle, the battle of the Indus was decided by a powerful charge by the keshig midday, the Seljuks were easily routed at Kose Dag in 1243, as with Dai Viet at No Nguyên in 1257, and the Xi Xia army was shattered in several battles with devastating charges in 1209 and 1227. The battle versus the Georgian army in the Mughan plain in 1221 may have been more drawn out and lengthy, as the Georgian army was surrounded and seems to have been slowly isolated and picked apart. A couple other battles where the Mongols were heavily outnumbered are known to have been long and drawn out: the Quyli River in 1219 and Parwan in 1211 being the two main examples that come to mind. Clearly the Mongols were consistently doing something to violently rout their foes, instead of wearing them down.
For equipment, Thomas of Spalato did state that Mongol bows pierced Hungarian armor, and that the reverse was not true. Hayton of Armenia noted that their shooting "commonly pierce all kinds of armour." Neither specified any kind of range where that was the case - some modern historians have estimated 50-200m, usually on the lower end, being the kill range. Still, it is notable that the only battle the Mongols routed the enemy just with firepower seems to have been at Mohi, where the Hungarians were trapped in a camp and may have been bombarded with incendiary gunpowder weapons. Muqali also used dismounted cavalry archers to break the impetus of a Jin attack in I believe 1219, but I'm not sure if firepower alone won the day. Otherwise, cavalry charges were an instrumental part of Mongol battles. Mongol bows also used a thumb ring, which seems to have been better than holding the arrow with 2 fingers as was common in Europe. Importantly, they used multiple types of arrows, presumably for different purposes and ranges. Carpini also noted that Mongols ideally were supposed to have had 2-3 bows along with other melee weapons, which the Liao Shi concurs with, specifying 9 pieces of armor and weapons.
As for tactics, Carpini mentions (from hearsay) that the Mongol subgroups would launch 3-4 waves of arrows before retreating back to their lines if resistance seemed tough. The tactics for ordo units in the Liao Shi indicate that the army attacked in a series of waves, with the lighter units firing arrows to pin and wear down the enemy and being replaced by units behind them to rest, while the heavier units waited until the enemy lines showed weakness. Chris Peers theorized that these waves were launched in echelon to prevent friendly units from running into each other. At the battle of Mount Yu in 1232, the Mongols attacked and retreated in 3 successive waves of fire, and then maneuvered their entire army behind the Jin position to attack their rear, mid-battle. The Jin managed to hold their ground and had a superior position, so neither side committed heavily. This noncomittal approach of advancing with fire and some light melee combat was also used against the Song in 1234 during their invasion of Henan: the Song managed to form squares/circles, so the Mongols did not commit to the attack and merely harassed them into starvation.
The best detailed firsthand account, to my knowledge, is from Babur's memoirs, which unfortunately is from some 300 years after Genghis Khan but with experience fighting in the same style in Central Asia. For the battles of Sar-I-Pul and Kandahar, he noted that both his men and horses were armored, his vanguard was assigned to men experienced with the sword, mentions arrow volleys/rains multiple times, and that spears were deployed in attacking a weakened part of the enemy line. The point of launching an all out melee charge when a section of the enemy formation looks ready to break is also mentioned in the report by Song envoy Peng Daya from his observations of the Mongols. Peng Daya also notes that 30% of the Mongol army consisted of heavily armored cavalry (though not all Mongol armies would have been uniform, particularly as the bulk of troops would have been from their subjects).
My hypothesis based on the scanty evidence of Song/Jurchen battles in the 12th century is that if the enemy army did not present a threat of quickly attacking, the Jurchen cavalry would circle up and deliver point blank shots with deadly impact or hack at the front lines with melee weapons, then wheel away for a new wave. The Song under Yue Fei managed to rout them by counterattacking with their zhanmadao blades, presumably because the more mobile Jurchen cavalry had already closed the distance either to fire point blank, harass with melee weapons, or commit to a charge.Ilkhanate/Mamluk battles have the best accounts for set piece action, and these describe the Mongols as attacking very aggressively, even when heavily outnumbered, and the Mamluks trying to absorb the Mongol combination of fire and shock, and counterattacking. However, its tricky to conclude too much because all but Ain Jalut and First Homs were post-dissolution armies, which may have fought differently.
The one intra-Mongol battle we have a good deal of info on is the battle of Herat in 1270, though this was between the post-dissolution Chaghatai and Ilkhanate armies. The Chaghatai army started the battle with a dense barrage of arrows, which apparently wounded a lot of people, and turned into a melee battle. The Chaghatai's heavily armored division attacked in multiple waves until the lkhanate left was routed. The Ilkhanate reserves, which included their heavy cavalry counterattacked, but it took three attacks before they succeeded - the Chaghatai division tried to retreat, but was surrounded by an ambushing Ilkhanate force that cut off its retreat back to the main body. The Chaghatai army supposedly lost 40,000 men, or most of their army, while the Ilkhanate lost around 5,000 of their larger army. The Ilkhanate arrows did manage to wound Chaghatai divisions, kill one of their key subcommanders, and loosen the enemy enough for melee attacks.
