Yes, this was tried in D&D 3.5e as a variant rule.
An officially published version of this rule existed in Unearthed Arcana, a 3rd edition D&D manual of variant rules. It can be used essentially unchanged with the 5th edition.
Unearthed Arcana p.133:
Players Roll All The Dice [...] Attacking and Defending
With this variant, PCs make their attacks just like they do in the standard rules. Their opponents, however, do not. Each time an enemy attacks a PC, the character's player rolls a defense check. If that defense check equals or exceeds the attack score of the enemy, the attack misses.
To determine the creature's attack score, add 11 to the creature's standard attack modifier (the number it would use, as either a bonus or a penalty to its attack roll, if it were attacking in any ordinary situation using the standard rules). For instance, an ogre has standard attack modifier of +8 with its greatclub. That means that its attack score is 19.
To make a defense check, roll 1d20 and add any modifiers that normally apply to your Armor Class (armor, size, deflection, and the like). This is effectively the same as rolling a d20, adding your total AC, and then subtracting 10.
- Attack Score 11 + enemy's attack bonus
- Defense Check 1d20 + character's AC modifiers
If a player rolls a natural 1 on a defense check, his character's opponent has scored a threat (just as if it had rolled a natural 20 on its attack roll).
Note that in D&D's core mechanic, a d20 roll that exactly matches its target DC is considered to succeed -- ties go to the person rolling the dice. So letting the defender roll actually gives them an advantage if you convert the numbers naïvely. This is why it's necessary to add 11 rather than 10 to the attack score to preserve the usual odds.
From the rules for Inspiration in the Player's Handbook, p.126:
If you have inspiration, you can expend it when you make an attack roll, saving throw, or ability check.
So you can't use inspiration to give yourself advantage on a defense roll. This matches the usual case, in which you can't use inspiration to give an opponent disadvantage on an attack roll.
Most everywhere I have seen they say RAW, it is DM discretion.
However according to Mike Mearls via Twitter (Twitter conversation), his advice is simply: Yes. They get death saving throws.
@tobyfernando: does the new BM Ranger companion get death saving throws upon reaching 0 hp?
@mikemearls: yes
**Please note that only Jeremy Crawford's tweets were ever official (but they no longer are). Tweets from Mike Mearls are treated as the personal rulings of a particularly well-informed DM.
Best Answer
I think that the common cold should be kind of trivial to adventurers whose constitutions are likely several standard deviations above the mean. In the context of D&D diseases, getting a cold seems more like an annoyance than anything else.
If you want, though, you could use the exhaustion mechanic (PHB291), which is a pre-existing mechanic that's well-suited to this. A single level of exhaustion gives disadvantage on ability checks, which might be consistent with a severe cold.