I see these examples (not just for the bard, but for all classes) as very good ways of showing how those classes exist in the world, what their roles are, and how they fit into the game from a narrative perspective.

That being said, the examples need not represent spellcasting at all. Not all spellcasting requires material components or a focus of some sort. In the examples provided, I can just as easily see those as representations of the bard's class features rather than the bard casting spells. I also don't know that the examples need to exactly match what can be done in game, for the same reason -- they're narrative examples, not gameplay examples. They might also be incomplete examples.

The first example is most likely a demonstration of the bard casting Legend Lore, which has V, S, and M components. The M components for this spell have a specific cost (250gp worth of incense and 4 ivory strips worth at least 50gp each), which means a bardic focus can't be used to cast it, so in this case, if she is casting Legend Lore (and it sounds like it, from the narrative description) there is no bardic focus.

The second and third examples seem to be narrative descriptions of various uses of the bardic class feature Bardic Inspiration which also doesn't require any sort of focus or musical instrument.

To address your specific concerns:

Is it reasonable to assume that RAW or at least RAI that the implication is that in the case of bards, they can use 1) their voice, 2) an improvised musical instrument, or 3) a bought musical instrument?

As far as RAW is concerned, you have one option: a musical instrument. I understand this is vague. The description in Chapter 5, Equipment (5e SRD) says:

Musical Instrument. Several of the most common types of musical instruments are shown on the table as examples. If you have proficiency with a given musical instrument, you can add your proficiency bonus to any ability checks you make to play music with the instrument. A bard can use a musical instrument as a spellcasting focus. Each type of musical instrument requires a separate proficiency.

The table in question is the Tools table, relevant section reproduced below:

\begin{array}{r|ccc} \text{Musical Instrument} & \text{Cost(gp)} & \text{Weight(lbs)}\\ \hline \text{Bagpipes} & 30 & 6 \\ \text{Drum} & 6 & 3 \\ \text{Dulcimer} & 25 & 10 \\ \text{Flute} & 2 & 1 \\ \text{Lute} & 35 & 2 \\ \text{Lyre} & 30 & 2 \\ \text{Horn} & 3 & 2 \\ \text{Panflute} & 12 & 2 \\ \text{Shawm} & 2 & 1 \\ \text{Viol} & 30 & 1 \\ \end{array}

This list is not meant to be exhaustive, of course, and your bard could use any musical instrument that exists in your world. However, the fact that playing a musical instrument has a specific entry in the rules says to me it's more than just banging on something and making noise. The fact the they require proficiency says to me that not just anyone can bang on a drum to make music. I also take it to mean that the bard requires proficiency in the instrument in order to use it as a bardic focus. This is not explicit but it stands to reason, since you need proficiency to 'use' a musical instrument, and a bard must use the instrument as a bardic focus -- this is the RAI part. I have a hard time believing that any designer intended to allow a bard to use a lute as a focus without being able to actually play the lute.

Taking all of this into consideration, I think it would be a stretch to allow a bard to bang on his armor, hum a few bars, or use some other type of improvised musical instrument, and without proficiency, use that as his bardic spellcasting focus.

1. Voice -- No. And even if so, you'd have a hard time saying your verbal components if you're also humming out a few bars as your arcane focus.

2. Improvised instrument -- No. You don't have proficiency in "banging on your armor."

3. A purchased instrument -- Yes. No problem there; it's in the rules.

And if such an assumption isn't RAW/RAI, what are the implications to allowing it as a house rule?

Not too much of a problem here. It's a cool narrative device. Spellcasting foci aren't required ("you can use a [thing] as a spellcasting focus" in every mention of foci in the class rules). It doesn't really affect the bard's ability to cast his spells whether he has a focus or not -- there's always the option of a component pouch -- and not all spells require foci. All in all if you want to allow it there's no real problem; in the end it doesn't make much of a difference.

Yes, you can; proficiency only adds your proficiency bonus to ability checks made to use your instrument

As you note, the bard's Spellcasting feature only says about instruments:

You can use a musical instrument [...] as a spellcasting focus for your bard spells.

This does not specify that you require proficiency with the instrument.

Just as a wizard can use any of the arcane focus options as a spellcasting focus, a bard can use any of the musical instrument options as a spellcasting focus.

The Tools section tells you what a tool proficiency does:

A tool helps you to do something you couldn't otherwise do, such as craft or repair an item, forge a document, or pick a lock. Your race, class, background, or feats give you proficiency with certain tools. Proficiency with a tool allows you to add your proficiency bonus to any ability check you make using that tool. Tool use is not tied to a single ability, since proficiency with a tool represents broader knowledge of its use. For example, the DM might ask you to make a Dexterity check to carve a fine detail with your woodcarver's tools, or a Strength check to make something out of particularly hard wood.

And as you quoted from the description of musical instrument proficiencies, they're only relevant to whether you can add your proficiency bonus to checks made to play the instrument.

Anyone proficient in an instrument (regardless of class) can add their prof. bonus to checks made to play it; and bards can use any instrument as a spellcasting focus regardless of whether they're proficient with it. The two uses are totally separate... though you may want to be proficient in the instrument you use as a focus if you're going to carry it around anyway.