Let's take these one at a time.
Making Death Saves is not the same as Dying
Let's look at the actual rules for dying...
Massive damage can kill you instantly. When damage reduces you to 0 hit points and there is damage remaining, you die if the remaining damage equals or exceeds your hit point maximum.
If damage reduces you to 0 hit points and fails to kill you, you fall unconscious. [...] Whenever you start your turn with 0 hit points, you must make a special saving throw, called a death saving throw, to determine whether you creep closer to death or hang onto life. [...] On your third success, you become stable (see below). On your third failure, you die.
As this rather clearly lays out, you "Die" only when you have failed 3 Death Saving Throws or taken Massive Damage. So, your Homunculous will die when you die, as laid out in the rules. Hitting 0 HP is not dead.
Homunculi can still be 'killed' normally
There are three general policies to keep in mind when examining D&D's rules.
- There are no 'secret rules,' if something does a thing--it will say so.
- Specific statements override general statements.
- When in doubt, use plain English to interpret a rule.
So, if we look at the Homunculous and the Create Homunculous spell, it says...
it dies if you die
It does not say "It dies only if you die." The actual statement is one-way causal... "If you die, then your homunculous dies, too." If it meant to say that your homunculous is functionally immortal as long as you're still alive, then it would say so.
This is roughly analogous to a statement like: "If you run your car into a tree, it will be damaged." Such a statement definitely does not imply that your car can only be damaged by running it into a tree.
Given the lack of a specific statement that says a Homunculous cannot be killed as long as its master lives, then the general rule of how things die applies to it. It just has an extra thing that can kill it, in the form of its master dying.