Yes, all of these examples work

Unless a class/domain feature says otherwise, they work. In fact the rules for multiclassing have a special mention for the Channel Divinity feature. The important part of that reads:

[...] Whenever you use the feature, you can choose any of the Channel Divinity effects available to you from your two classes.

This is specifically referring to gaining the Channel Divinity feature from multiple sources, but also means that the effects can work on qualifying spells/attacks from either class.

Specifically for your examples

Channel Divinity: Destructive Wrath on shocking grasp

The Tempest cleric's Destructive Wrath Channel Divinity option (PHB, p. 63) says:

When you roll lightning or thunder damage, you can use your Channel Divinity to deal maximum damage, instead of rolling.

A successful attack with shocking grasp deals lightning damage; this qualifies for your Channel Divinity feature. There is no restriction on the source of the spell.

Divine Strike on booming blade attacks

The Tempest cleric's 8th-level Divine Strike feature says:

Once on each of your turns when you hit a creature with a weapon attack, you can cause the attack to deal an extra 1d8 radiant damage to the target.

Booming blade says:

As part of the action used to cast this spell, you must make a melee attack with a weapon against one creature within the spell’s range, otherwise the spell fails. On a hit

Since booming blade requires a weapon attack, you can use Divine Strike on it.

Arcane Ward recharging from freedom of movement

The Abjuration wizard's Arcane Ward feature (PHB, p. 115) triggers:

When you cast an abjuration spell of 1st level or higher

Freedom of movement is an abjuration spell and will refill the ward.

Sculpt Spell and Channel Divinity: Destructive Wrath on shatter

The Evocation wizard's Sculpt Spell feature states:

When you cast an evocation spell that affects other creatures that you can see [...]

Shatter is an evocation spell, so you can use Sculpt Spell; it deals thunder damage, so you can use Destructive Wrath. This works.

Examples that don't work

Light Domain cleric's 8th level ability, Potent Spellcasting states:

... you add your Wisdom modifier to the damage you deal with any cleric cantrip.

As this ability specifically calls out that the cantrip must be a cleric spell it will not work with your multiclassed cantrips.

_{Thanks to MivaScott for the example}

Other considerations

Unlike previous editions, 5th edition makes no distinction between the sources of your spell slots for multiclassed spellcasters, with the slight exception of warlocks as Pact Magic works differently. However, even for warlocks, you can cast any prepared/known spell from any class in any slot of the appropriate level.

To use your class features with these spells, they just have to meet the requirements specified in the ability/feature. There isn't even a distinction made for the source of the spell. You can apply class features to spells cast from magical items.

With only a pact weapon, you can only make two attacks

Thirsting Blade says:

You can attack with your pact weapon twice, instead of once, whenever you take the Attack action on your turn.

And Arcane Armament says:

Starting at 5th level, you can attack twice, rather than once, whenever you take the Attack action on your turn, but one of the attacks must be made with a magic weapon, the magic of which you use to propel the attack.

Note that Thirsting Blade does not say that you get to attack "one extra time" with your pact weapon. It says you get to attack twice when you take the Attack action with that weapon. Thus, if you are using only that weapon to make your attacks, you will only be able to attack with it twice per Attack action. Arcane Armament just adds a different way to achieve two attacks - but neither ability will give you a third attack in this situation.

What if you make one attack with a weapon that is not your pact weapon?

Take this example: you made one attack with a non-pact weapon, dropped it then made the second attack of arcane armament with the pact weapon.¹

In this case, it actually creates a slightly ambiguous situation if you read it strictly by RAW.

By RAW, Thirsting Blade allows you to attack twice with your pact weapon specifically, and Arcane Armament allows you to attack twice in general; both effects trigger off of the above situation. If an ability is triggered, you should be able to benefit from its effects unless they are prevented by something else. The only way for both of those abilities' effect descriptions to be accurate would technically be to allow another attack with the pact weapon. And there is seemingly no rule preventing this from happening.

However, this is almost certainly not the intent at all. The fact that Thirsting Blade explicitly was added as an exception to Extra Attack (which is almost identical to Arcane Armament) tells us that the kind of interaction is not intended. Note that the artificer is in UA, and not all UA is balanced or tweaked for multiclassing. This is likely just a case of a loophole they haven't gotten around to addressing or noticing yet, but I'd expect it to be fixed upon official release.

^{1 - Two-weapon fighting would not work here, however, since the bonus-action attack is taken after the Attack action and not as part of it.}