Yes

Many parts of the text on attunement specify that they only apply to items that require attunement. Surprisingly, actually attuning isn't one of those parts. The DMG ( p. 138) states:

Attuning to an item requires a creature to spend a short rest focused on only that item while being in physical contact with it (this can't be the same short rest used to learn the item's properties)..., at the end of the short rest, the creature gains an intuitive understanding of how to activate any magical properties of the item, including any necessary command words.

None of this demands that the item in question requires attunement.

Also note that you must spend a different short rest (ie, another hour) to identify what the properties of this magic item are (attunement does not reveal that). And yet another hour to drop attunement to this item, since you can only be attuned to up to three magic items at a time. Still, you will be able to learn the number of charges an item has without the Identify spell.

NOTE: Spending a short rest to Identify the item (DMG p. 136) will not reveal how many charges the item has left, by rules as written (RAW). Your DM may permit it to do so, but RAW states that the Identify spell reveals charges remaining (both stated on DMG p. 141, and PHB p. 252, in the text of the spell), while the description of mundanely identifying the item (with a short rest) (DMG p. 136) does not.

As you have noted, it's unclear. That means it's DM's call.

There are a large number of points in the rules of 5e that are not entirely clear. It is explicitly intended that the DM adjudicate those things at their table. It's just one of the features of the game system - they're actively encouraging table adjudications and houserules.

That having been said...

This is in a published book. We can make a reasonable assumption that the wording is as intended... and the wording doesn't say. If we look at the rules for Bardic Inspiration, we see the following (emphasis mine):

Once within the next 10 minutes, the creature can roll the die and add the number rolled to one ability check, attack roll, or saving throw it makes. The creature can wait until after it rolls the d20 before deciding to use the Bardic Inspiration die, but must decide before the DM says whether the roll succeeds or fails. Once the Bardic Inspiration die is rolled, it is lost. A creature can have only one Bardic Inspiration die at a time.

The wording here states outright that in this particular case, the player must decide before the DM says whether the roll succeeds or fails. By implication, then, that's not inherent in the fact that it's giving a bonus to the roll.

As such, it should be possible to give a bonus to a roll after success or failure has been determined - and, indeed, the Divine Soul sorcerer's Favored by the Gods feature (XGtE, p. 50) does this - and Flash of Genius leaves activation in the hands of the player.

Thus, it would make sense that the player could wait until after success or failure had been declared before deciding whether or not to use Flash of Genius.

eidt: and, we now have new information... sort of. The Sage Advice Compendium offers a not-entirely-clear clarification to go with the above.

You use Flash of Genius immediately after the triggering d20 is rolled and before any of the effects of the roll are applied. Unless a rule tells you otherwise, a reaction occurs immediately after its trigger

So... it's after the roll, but before the effects are applied. Yes, that seems to cover both of the previously described options. Admittedly, it does tend to nudge it closer to the "probably before success/failure is declared" side of things, but to my eyes it's still firmly within DM adjudication land.