Attunement is not alignment restricted unless the specific item says it is (and this item does not say that)

There are no rules about who can attune to sentient magic items specifically, so the normal rules apply. If the item does not say that only a specific alignment can/can't attune to it, then any alignment can attune to it.

One example of this (as pointed out in Medix2's answer) is the Moonblade, which says:

(Requires Attunement by an Elf or Half-Elf of Neutral Good Alignment)

Hazirawn has nothing in it that would suggest any kind of alignment limitation on it, so any alignment can attune to it.

When the user conflicts with the item, the item can make demands

Now, as far as what happens when a user with a conflicting alignment does attune to it, the rules for sentient magic items do go into some detail:

A sentient item has a will of its own, shaped by its personality and alignment. If its wielder acts in a manner opposed to the item's alignment or purpose, conflict can arise. When such a conflict occurs, the item makes a Charisma check contested by the wielder's Charisma check. If the item wins the contest, it makes one or more of the following demands [list omitted]

So, while not explicitly limited to alignment based conflicts, it seems entirely likely that a user weilding a weapon that is in opposition to their alignment would come into conflict with that weapon's special purpose (if it has one) at some point and the rules describe how to deal with it (up to and including the item taking control of the user).

The Hoard of the Dragon Queen does not note any particular goals or behaviors of Hazirawn, but a DM can (and likely would) come up with some things to make it interesting.

Yes, creatures created by Animate Dead can use wands and other magic items.

The rules regarding Zombies and Skeletons don't say that they can't be instructed to use magic items. They also don't say they can't attune to items, so we have to revert to the general rules on attunement.

A magic item’s description explains how the item works. General rules on using magic items are deliberately scant in 5th edition, deferring to the specific rules for each item. But in general unless otherwise specified, any creature can use a magic item.

For example the Wand of Magic Missiles just requires a creature to hold it and use an action. Zombies and Skeletons can accomplish this since they are creatures, they can hold objects, and they can take actions.

This includes items that require attunement. The rules on attunement in the Dungeon Master's Guide specify the requirements for attuning to magic items:

Attuning to an item requires a creature to spend a short rest focused on only that item while being in physical contact with it.

Then:

... at the end of the short rest, the creature gains an intuitive understanding of how to activate any magical properties of the item, including any necessary command words.

Zombies and Skeletons are creatures and can take short rests, so they can attune to magic items unless the description of the item says they can't.

Of course, neither of these creature types can use magic items that have a command word, since the description of both the Zombie and the Skeleton says that they "can't speak".