I am pretty sure that the rules never clarify beyond what you have already quoted, and I am certain that the rules you have quoted are not definitive on this point.

I, and literally everyone I have ever played with or talked to, have always assumed that casting spells with both material and somatic components takes one hand. Never really paid very much attention to it. But I suppose the assumption was either that the material component is consumed before the gesturing takes place, or else that the gesturing is done with the hand holding the material component.

Yes, it can

You can make any non-magical trinket that can fit in your hand, and it disappears after the end of your next turn. One of the trinkets you can make, as listed under PHB 160, is "a whistle made from gold-colored wood" as well as "a tiny silver bell without a clapper"; these are both things that you could make such that they can fit in your hand.

Furthermore, they fit the criteria of "musical instruments," because you can make music with these. While the PHB does list some of the things it considers as instruments, it is not meant to be an exhaustive list:

Tools, PHB 154

Musical Instrument. Several of the most common types of musical instruments are shown on the table as examples.

This means prestidigitation can create a musical instrument that can fit in your hand, and you can use any musical instrument as a spellcasting focus.

Of course, it is inefficient

You must use your action to cast prestidigitation, and then you cannot use your action again to cast another spell. On your next turn, you could use the instrument to cast a spell, but then it disappears at the end of that turn, and you have to recreate it on your third turn. This allows you one spell every other round that isn't prestidigitation (unless it's a bonus action or reaction spell).