We actually get our first mention of rounds and turns in Chapter 8: Adventuring;
In combat and other fast-paced situations, the game relies on rounds,
a 6-second span of time.
You'll note that the game explicitly states that rounds are both for combat and "fast-paced" situations. This seems open and shut. Rounds are for combat, yes, but also fast-paced situations.
However, this "fast-paced" situations line appears to be a throwaway as it is never referenced again throughout the entire rules. If you continue reading the rules you will find the mechanics for rounds listed in Chapter 9: Combat. The section The Order Of Combat in Chapter 9 opens by explaining that rounds and turns are a method of organizing combat;
A typical combat encounter is a clash between two sides, a flurry of weapon swings, feints, parries, footwork, and spellcasting. The game organizes the chaos of combat into a cycle of rounds and turns. A round represents about 6 seconds in the game world. During a round, each participant in a battle takes a turn. The order of turns is determined at the beginning of a combat encounter, when everyone rolls initiative. Once everyone has taken a turn, the fight continues to the next round if neither side has defeated the other.
You can see that a lot of the language used explicitly talks about fighting; "each participant in a battle takes a turn", "The order of turns is determined at the beginning of a combat encounter", "the fight continues to the next round".
Some other chapters make mention of rounds, but only in the context of combat (chapter 11, 12, 13, and 14, plus the classes of course).
Rounds as a framework for non-combat fast-paced activities
Looking at other texts, we see the same pattern. For example the DMG talks about initiative, but again explicitly says they are talking about combat; "You can use several different methods for keeping track of who goes when in combat".
However in the DMG we do get our only non-combat use of rounds (albeit, as a suggestion for a table ruling) that I could find in any source book: Chases.
A chase requires a quarry and at least one pursuer. Any participants
not already in initiative order must roll initiative. As in combat,
each participant in the chase can take one action and move on its
turn. The chase ends when one side drops out or the quarry escapes.
When a chase begins, determine the starting distance between the
quarry and the pursuers. Track the distance between them, and
designate the pursuer closest to the quarry as the lead. The lead
pursuer might change from round to round.
So are there rounds during a short rest?
The answer is by pure RAW, no. The only time that you enter the rounds-turn structure is in combat by RAW. Everything else is up to DM ruling - and to be honest "when to enter combat" is also ill-defined and up to DM ruling.
The rules present the rounds-turns structure as a tool for DMs to use, and the DMG presents an example of using them for chases. While this isn't "the rules", it is "a ruling". In the event of simultaneous actions taking place out of combat, the DM has to resolve them somehow. I don't think that the rounds-turns structure is an ill-fitting tool, however it is one with a lot of overhead.
My experience using rounds outside of combat
I have done it before, for conversations between multiple people. In my opinion, it's not worth it. Entering rounds is a heavy-duty tool, it takes time to set up, it slows the game down, it restricts the free-flowing nature of the game to a static structure, it doesn't integrate well with multiple people doing different things.
If two people declare they do something at the same time, they simply roll an opposed initiative check, and then we are done with it. I have always found this simplest, easiest, and doesn't get in the way of gameplay.
Can you simultaneously end a long and short rest?
Firstly, short and long rests are both things characters do. They have no defined end, besides the characters stopping resting. In my experience that's how it works at most tables too, players simply declare they are going to stop resting and get on with their day, and so they do.
You should be aware that both have different things that stop them. For example fighting or casting spells will end a short rest, but not a long rest. If you want to conclusively end your short rest before your long, that's quite easy to do.
There is no further guidance given as to when rests end. As strange as it is, you could adventure for nearly an hour before receiving the benefits to a long rest, and that would be fine rules wise.