I play a D&D 5e campaign. My Tempest Cleric is about to take his first multi-class dip into Wizard.

Am I able to use my Cleric class and domain features on my Wizard spells, and vice-versa? Both at the same time?

Examples:

• When I cast Shocking Grasp (Wizard spell), can I use my Channel
  Divinity: Destructive Wrath to maximize the lightning damage and then
  use Thunderbolt Strike (another domain feature) to cause the target
  to be pushed back 10 feet?


• When I roll damage on the weapon attack made for Booming Blade
  (Wizard spell), can I also add the 1d8 or 2d8 Thunder damage from my
  domain’s Divine Strike feature to the total?


• If I study the School of Abjuration, is my Arcane Ward replenished if
  I cast Freedom of Movement (Cleric abjuration spell, not on Wizard
  spell list)?


• If I study the School of Evocation, can I cast Shatter (Cleric domain
  spell) and apply both Sculp Spell (Wizard School of Evocation
  feature) to avoid hitting my allies in the blast radius and ALSO use
  Channel Divinity: Destructive Wrath to maximize the thunder damage?

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.

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.

All your examples work

but only because the features you mentioned does not restrict its effect to a specific class.

1. Channel Divinity: Destructive Wrath and Thunderbolt Smite only requires you to deal lightning or thunder damage and lightning damage, respectively, with no mention of the source or class.


2. 
   Booming blade tells you to make a weapon attack. Divine Strike only requires you to hit a creature with a weapon attack. You can apply Divine Strike.


3. Arcane Ward only requires you to cast level 1 or higher abjuration spell. Freedom of movement is an abjuration spell, cast at level 1. You can use Arcane Ward.


4. Sculpt Spell only requires the spell cast is evocation. Channel Divinity: Destructive Wrath only requires the damage dealt is lightning or thunder damage. Both work independently.