The Detect Magic spell states:
If the aura emanates from a magic item, you can attempt to identify its properties (see Spellcraft).
Spellcraft states:
Identify the properties of a magic item using detect magic: DC 15 + Item's CL
So what, you might ask, is a "property?" Well, they don't define in a legalese way exactly what is included in a magic item's properties, except to note that it definitely gets you command words ("The spells detect magic, identify, and analyze dweomer all reveal command words if the properties of the item are successfully identified..."). One simply has to assume from general English definition and logic that it gives you anything beyond that, including what the item even does.
I think it's best to interpret "properties" as "All of what it does, including charges, command words, and whatnot. Its full rules stat block." (Excepting, of course, other defined exceptions like artifacts and spells on a scroll.) Analyze dweomer specifically says it gets charges, but relying on a 6th level spell to get the charge level of a plain old wand is pretty lame IMO.
In earlier editions of D&D I was fine with not telling people charges and letting them find out when they ran out - it added a nice randomization factor - but in Pathfinder where the Christmas tree syndrome tends to dictate that it's players' God Given Right to liquidate all treasure for a union-decreed cost to buy other gear, not knowing charges and thus value would be an impediment.
It should be fine, as long as time allows.
Nothing in the rules says that a long rest cannot immediately follow a short rest. In this case, what the characters are doing during those rests are very different.
Indeed, a short rest and a long rest are what many real-life people do before bed: an hour or so of non-strenuous activity, like reading, and then going to sleep. It's not a stretch to say that a D&D character can't spend an hour studying their magical item, and then go to sleep.
The only time where that might be a problem is if the characters don't have the full 9 hours.
Different things are happening during the two rests.
What's really key here is that the activities during the two rests are different. In the short rest portion, the character is studying the magic item, meditating on it, or whatever is required for attunement. In the long rest portion, the character is doing something else, such as sleeping. The attunement process is still active work, just not active relative to adventuring.
I think that this distinction is why attunement is limited to short rests, and why a character should be able to do chain them together.
Best Answer
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:
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.