614.6 If an event is replaced, it never happens. A modified event occurs instead, which may in turn trigger abilities. Note that the modified event may contain instructions that can’t be carried out, in which case the impossible instruction is simply ignored.
If a Commander would be put into a library, hand, graveyard or exile from anywhere, its owner may choose to move it to the command zone instead. Details
This is a replacement effect. It applies last and may apply multiple times to an event.
The creature never goes to the original destination zone and will not trigger abilities that trigger based on going there (e.g. "dies" triggers).
rezombied is incorrect; you will get the mill effect even if you put Lazav in the command zone. 614.6 has no bearing on this interaction. The rule you're looking for is this one, bolding mine:
117.12. Some spells, activated abilities, and triggered abilities read, "[Do something]. If [a player] [does, doesn't, or can't], [effect]." or "[A player] may [do something]. If [that player] [does, doesn't, or can't], [effect]." The action [do something] is a cost, paid when the spell or ability resolves. The "If [a player] [does, doesn't, or can't]" clause checks whether the player chose to pay an optional cost or started to pay a mandatory cost, regardless of what events actually occurred.
It doesn't matter that the event of shuffling Lazav in didn't happen, that's not actually what the ability is looking for. It only needs you to pay the mandatory cost of initiating the first action it tells you to do, which does happen even if the event that that cost causes is modified (otherwise there would be nothing removing Lazav from the battlefield and allowing you to put him in the command zone).
The "if you do" part is only there for the eventuality that the Phantasm would be removed from the battlefield before the ability resolves, then of course it doesn't work.
I'm a former judge (lapsed), who keeps up to date on rules and policy. Keep in mind that judges' answers aren't necessarily more valid than those of people who aren't judges; what matters is we can quote the rules to back up our answers. When in doubt, ask for such quotes.
Note that when Mirror-Mad Phantasm's activated ability resolves, Mirror-Mad Phantasm's owner shuffles their library "even if ... an effect", such as the effect imposed by C.R. 903.9, "causes [Mirror-Mad Phantasm] to be moved to another zone" (C.R. 701.19c).
Of course, for this particular interaction, even if you did have to put Lazav in the library in order for it to work it wouldn't matter; you're milling your entire library, including the Lazav you shuffled in, at which point you would again have the option of putting him in your command zone.