Attuning does not interrupt a short rest
Under the attunement rules, it says:
If the short rest is interrupted, the attunement attempt fails.
If the process of attuning to the item itself is enough to interrupt a short rest (ie, because attuning is not "light activity" as required by resting), then the item remains unattuned after the short rest as well, as stated in the quoted rule.
This means nobody can attune to anything because attuning interrupts itself: it is non-light activity you must take over a period of time that requires light activity (ie, it's a contradiction, this rule is nonsensical).
So we must conclude that attuning to an item does not interrupt the short rest, and therefore attuning to an item must fall under light activity, as otherwise the design of this mechanic prevents anybody from using it.
If the short rest is not interrupted, they gain the benefits of a short rest
If attuning to an item does not interrupt a short rest, and nothing else interrupts it as well, then a PC can gain the benefits of such a rest. This includes recharging abilities, using features that key off short rests (eg, Inspiring Leader feat, Bard's Song of Rest), and, yes, using Hit Die to heal HP.
There are no disadvantages listed while attuning to an item
Focusing on an item during attunement does not impose disadvantage to Perception checks while they are on the lookout during a long rest. There is no mention of this in the DMG. But, the DM is free to impose any disadvantages he feels necessary as part of the focus attuning requires.
Impact to the game
I feel that if my DM imposed this penalty on me, I would just ask the members of my party to attune to items over short rests only, and to never do it during long rests. And then this penalty would never come into play, which makes it, on the whole, an immaterial penalty (unless an adventure is specifically designed with this in mind).
Options from 5th Edition
- In the PHB there is a Spell called Bestow Curse on pg 218 with some options.
- The Hex Spell PHB pg 251 (not a lot of options there though)
- If you use the scroll and potion mishaps, you will want to remove the beneficial effects.
- You could also consider some effects from the Wild Magic Surge table on pg 104 of the PHB
- The DMG pg 220 has a list of Detrimental Properties for artifacts that can be used for curses.
- There is a list of Vistani Curses on page 28 of Curse of Strahd. Curse of Strahd is a good resource for ideas on curses, and cursed items.
Options from Earlier Editions
Since these types of effects tend to be more edition agnostic than other rules, you might want to consider earlier sources. Dragon Magazine would often run an article on new ways to curse a player.
- Dragon 348 pg 34 Article: Bestowed Curses
- Dragon 229 pg 26 Article: Curses
- Dragon 34 pg 30 Article: Quirks and Curses
- Dragon 167 pg 28 Article: Curses Are Divine
- Dragon 77 pg 18 Article: Curses! (by Ed Greenwood)
- Dragon 229 pg 26 Article: Curses
- Dragon 29 pg 13 Article: Curses: Never Get Even - Get Ahead!
- Apparently 3.5's Book Of Vile Darkness contained some suggestions, but I can't get my hands on that book right now to give a page reference.
- The 2nd Edition DMG Under Magic Items: Scrolls: Cursed Scrolls has a list of great suggestions, some of which my characters have been subjected to. (aka how my elf got a beard.)
- The 3.0 DMG has a list of interesting scroll mishaps under Magic Items: Scrolls: Scroll Mishaps on pg 203.
- The 3.5 DMG has a bunch of tables to play with on pages 272 - 273.
- Every edition of the DMG has some sort of list of drawbacks to place on artifacts that can be used for curses.
While it's not entirely related, here is a fun article on ways players can use cursed items to their advantage: "Putting Evil To Good Use" by Jonathan M. Richards in Dragon 215 pg 10
I am sure there are many more sources out there, so I encourage you to keep exploring.
General Suggestion
Rolling randomly is OK if it is a scroll / potion / trap, or even a magical mishap. But you might want to consider tying the curse to to the theme of the item when the item is more significant. Look in the DMG at the cursed items there for inspiration, and let your imagination run wild!
Because a cursed monocle that makes you half your height is forgettable, but one that makes you talk and act like a posh Waterdhavian Noble with a heap of contempt for their 'lessers' is much more fun.
Best Answer
I'm pretty sure the answer to your question is "no". The ring of sustenance first appeared in Unearthed Arcana (1985), and the description did not change until it disappeared from the game after 3.5e.
Some speculation on your questions: