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.
