What is the dependent clause in these sentences and are they nouns, adverbs, or adjectives?

All family members know why the dog must leave.

Why the dog must leave? Adverb

They realize that the dog will become a visually challenged person's constant companion.

They realize that? Noun

You've identified the first dependent clause -- but it is not an adverb. It answers WHAT after the verb.

In the second sentence, the clause must include the verb and complements. Yes, it's a noun clause.

The first is an adjective.

The second is they realize that the dog will become?

Direct objects answer WHAT after a verb.

You still don't have the complete dependent clause in the second sentence.

The first is a noun.

The second still confuses me because I have to leave an independent clause after naming the dependent clause....

Yes. The first one is a noun clause.

Independent clause: They realize

Dependent clause: that the dog will become a visually challenged person's constant companion.

One more...

In fact, a visually challenged person can handle most traveling situations if a guide dog is with him or her.

If a guide dog is with him or her?
Adverb?

Yes, you're right. :-)