The short answer: yes, you must have identified a magic item before you can attune to that item.
To address your questions directly:
1. 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 it's properties, especially if you don't know that it requires attunement. The rules specifically require that the short rest used to attune be a different short rest than the short rest used to identify its properties:
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).
5e SRD, Magic Item Attunement
The DMG also has this to say about identifying magic items (paraphrasing):
If a player has just found a magic item and can’t cast the identify
spell, they must spend a short rest to determine what the magic item
is. Potions are an exception, just a little taste of the potion is
enough to tell you what the potion does.
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.
2. If it is usable without attunement and 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.
3. 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.
4. 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 missile 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.