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.
Interpreting the language of descriptions is left to the DM. To me it is clear that the intention was to hinder any divination spell that reveals information about magical properties, but this is my interpretation, so check with your DM.
Due to the magic aura, identify will show obfuscated or false information, but it will also detect that the item is under the magic aura spell. While you know that the information might be falsified, you cannot identify its true properties until you get rid of the aura (with dispel magic or letting its duration end).
The spell explicitly states that you can change whether the item appeares magical or not. Beyond that you can make it show that the magic belongs to a specific school of magic. This is mainly designed to fool detect magic. Because identify reveals the magic aura spell anyway, showing a specific ability or effect would be pointless.
Best Answer
The short answer: yes, you must have identified a magic item before you can attune to that item.
To address your questions directly:
If it's attunable, can you attune to it without having spent a short rest to learn its properties? Thus requiring only one short rest.
No. You can't attune to a magic item if you don't understand its properties, especially if you don't know that it requires attunement. The rules specifically require that the short rest used for attunement be a different short rest than the short rest used to identify its properties (emphasis mine):
The DMG also has this to say about identifying magic items:
Since we know that the same short rest can't be used to both identify and and attune to an item, and we don't know that an item requires attunement until after it has been identified, it stands to reason that you can't attune or otherwise use an unidentified item.
If it is usable without attunement and doesn't require a command word, like a Wand of Magic Missiles, can you use it?
My gut says not until you know its properties and how it works (which you probably don't if you just picked it up). If you have seen such an object before then it's possible you already know what it is and how to use it, and in you haven't seen such an object before your DM may allow you to experiment with the object; he might also not. This is all squarely in the realm of "ask your DM how this works." Expect table variation.
If it is usable without attunement but requires a command word, is it enough if you saw an enemy or ally use it once?
Maybe. This really depends on your DM. I might say yes, but I also might make it blow up in your face, depending on your character's knowledge of the item and his familiarity with magic and magic items. In all cases, expect results to vary from table to table.
Is it enough if an ally of yours knows the properties of the magic item and tells you about it to be able to use/attune to it? How long does this process take? I imagine if it took you 1 hour to identify the properties, it might take some time to explain them to others, right?
This is also situational. If it's a complex magic item with many functions it's less likely to be explained easily. On the other hand, if it's a wand of magic missiles and I know how to use it ("Direct pointy end at target, wave emphatically") then I should probably be able to explain that pretty succinctly to my fellow party members. I would say to this also, expect table variation.