Spiritual Weapon (Hammer) can not be twin cast

It doesn't target a creature at all, so it is not twinnable.

Dominate Beast is eligible, and presents a sticky situation

As you've quoted, the language of dominate beast states:

You can use your action to take total and precise control of the target.

Spell is twinned for two targets, but the spell language doesn't change

Once twinned, you now have two beasts under your control, but unless you have a means for a second action, you only have one action available to control one beast. The twinning of the spell only allows you to target two creatures, it does not change the wording of the spell used.

This means you are still limited to controlling one beast with your action.

The argument for allowing it

The other side of this is that with Twin, you aren't casting two spells, but just one spell that has two targets. Anything you do afterwards would be twinned as well and thus a single action would be sent to both beasts.

So...DM call

Unfortunately, this does seem like it can be a DM decision and what feels right at your table. There is reasonable logic for each case.

Probably not

You select the Twinned Spell target at the same time you select your primary target for the spell from War Caster. It says "When" you cast the spell. In other words, there are not two rounds of target selection, only one.

You can cast a spell that does not require an attack roll, but I think looking at the rules for making an attack helps reinforce this interpretation:

1. Choose a target. Pick a target within your attack's range: a creature, an object, or a location.

2. Determine modifiers. The DM determines whether the target has cover and whether you have advantage or disadvantage against the target. In addition, spells, special abilities, and other effects can apply penalties or bonuses to your attack roll.

3. Resolve the attack [...]

Set aside the question of whether or not twinning is allowed for a second. The reaction is resolved by first declaring targets (at this point, the caster would pay the sorcery point(s) to add a second target if it were allowed), determine what sort of bonuses or penalties each roll would make, and then roll the dice to make the spell attack.

Obviously, the spell might not involve an attack roll, in which case the latter steps are skipped (or they apply to the target if it needs to make a save).

But we see only one round of target selection because it's one spell. Twinning just gives the caster the option of expanding that selection to more than one target, but War Caster precludes selecting more than one target for the spellcasting. All this is established before the spell's effect is resolved and the casting is finished.

So I think the rules indicate that one can't twin a spell cast using the War Caster feat.