I was trying to visualise how Mongol cavlary could have holstered lances (vertically?) to use a bow when I realised that nothing I had ever read actually stated that all Cavalry carried bows, and that I was just assuming that they would.
I can comment on the second question!
I had the same question. /u/SlavicThunder points out that the 'Avar-style' lance had a leather thong, and the passage says 'some...have a spear also attached to them,' in addition to the bow.
I have a couple images that might show how that actually worked. They're of much later-period horsemen, but the method can't be that far off from what a Mongol might've used: steppe cavalry carrying their lances on a shoulder strap on their draw arm.
Notice how lightweight (by Western standards) those lances must be, even accounting for archers' upper body strength.
