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

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. :-)