No, short rests and long rests are mutually exclusive.
The PHB has the following to say about long rests:
If the rest is interrupted by a period of strenuous activity - at least 1 hour of walking, fighting, casting spells, or similar adventuring activity - the characters must begin the rest again to gain any benefit from it.
From this we can infer that if you are trying to take a long rest, you get no benefits from it until you complete the full rest. If sufficiently interrupted partway through - no matter how long you had been resting beforehand - you must begin the rest again to gain any benefit from it. There is no allowance here for getting the benefits of a short rest if you managed to go at least an hour before getting interrupted.
As written, you would have to first choose to have a short rest, wait out that hour, and then begin a separate long rest if you wanted to try and get the long rest's benefits. Luckily, if the rest interruption lasts less than an hour - which would certainly be the case for most combats - you can finish the current rest and get the full benefit at the end.
V2Blast's answer includes reference to comments from the designers which clarify that these are the rules as intended, or at least as subsequently interpreted by their writers.
I personally am not much a fan of this interpretation. A rest is a rest, and it doesn't make a lot of sense that how you feel after resting for an hour depends on whether you planned to only rest for an hour or were trying for a full night. Additionally, there is no requirement stated in the rules that the party must declare the kind of rest they are trying to have when they start to do so (the necessity to do so is only implied by the way that the designers think rests should work).
I would personally rule that: if you sit down to rest and make it at least an hour before being interrupted, you get the benefits of a short rest, and then if you continue resting to the full eight hours (and get your six hours of sleep), you get the benefits of a long rest.
Though you'd probably prepare a little differently if deliberately settling down for a long rest as opposed to a short one, the actual activity restrictions during short or long rests are almost identical save for that a long rest requires you to get six hours of sleep in the eight hours total - that leaves two hours in which you can do exactly what you would have been doing if you were taking a short rest, so they hardly seem exclusive to me. If you enforce separation between long and short rests, you're likely to have awkward logistics where wary players always have a short rest and then have a long rest, in order to get the short rest's benefits just in case they are interrupted during the night - and this mechanical complexity seems unnecessary and unrealistic. Just let a character rest!
It is important to note that this covers one continuous resting period (as the rules say, long and short rests are periods of "at least" the specified time) and so I would say you should only be able to take the benefits of a short rest once, after the first hour of the rest period; if you allow every hour of an eight-hour rest to count as an individual short rest as well as being a part of a long rest, you will probably run into some issues where players can abuse the mechanics.
According to the spell description, that wouldn't work.
If you enter a new plane or return to the plane you were on when casting this spell, your body and possessions are transported along the silver cord, allowing you to re-enter your body as you enter the new plane.
The moment you return to the Material Plane, your body snaps to you- you can't reside eternally on the Material Plane in your astral form.
While you're in your astral form, you don't age or feel the effects of hunger, but no other effects are given, save one:
Since the Outer Planes are as much spiritual states of being as they are physical places, this allows a character to manifest in an Outer Plane as if he or she had physically travelled there, but as in a dream.
A character's death- either in the Astral Plane or the destination plane- causes no actual harm.
You could thus stash your body in a closet, astrally project, and become immortal in the Astral Plane or Outer Planes- or you could teleport about the Material Plane, Ethereal Plane and Inner Planes by means of astral projection, but only in the Astral and Outer Planes do you have the protection inherent to astral projection- most likely because those are the planes where you do not exist in a wholly physical form.
As for resting during astral travel:
It should be noted that several features of the Outer Planes take effect when you complete a long rest (notably psychic dissonance), and nothing about the Astral Planes says that you cannot take a rest, so you can indeed take long or short rests in astral form. In order to be kept from taking those rests, the spell description would need to specifically prevent you from doing so.
Resources are regained at the end of a long rest.
The barbarian's rage class feature states:
Spellcaster's spell slot section states:
For hit points and hit die:
So it doesn't matter what is spent during the rest as long as the PCs get to finish that rest. This also means that any resources spent before the rest started remain spent during the rest and ensuing combat, until the long rest is completed.
It's also worth noting that it's not intended for the PCs to be able to gain the benefits of a short rest during a long rest, as per this unofficial guidance by rules designer Jeremy Crawford on Twitter:
Obviously a DM can rule whichever way they want but the designers didn't intend for a long rest to include a short rest.