1. the teacher who teaches us
2. the teacher teaching us
In the first example, "who teaches us" is a relative clause that restrictively modifies the noun "teacher," thus adjectivally. The second example means what the first example means, but it is not a reconstruction or a version of the first as it is employing a participial phrase as a post-positive adjective.
A postpositive adjective is a predicative adjective that is placed after the noun or pronoun that it modifies. This contrasts with prepositive adjectives, which come before the noun or pronoun.
One type of postpostive adjective is participles. The phrase "teaching us" is a participle, "teaching" being the present participle of the verb "to teach." As a participle, it can be placed in a post-positive position of the noun, or after the noun.
1. the man who is happy
2. the man happy
3. the happy man
Like before, neither the second nor third examples are versions of the first. So be clear on that. Also, you seem to be under the misguided perception the second one is wrong. It's not. While, "happy" is generally not a postpostive adjective, as per the link above, it is sometimes, for example:
1. I want to see the man happy.
2. Make the man happy.
In the above, the reason for the post-postive placement is "the man" is being used pronounally for a specific person. Referring to the the link above, you will see that when we modify pronouns, the adjectives appear postpostively.
"Kennedy is the person generous to everyone."
That's kind of a weird sentence, but it's not ungrammatical. If you review the link on post-positive adjectives, you will see a section that discusses post-positive placement when using "an adjective phrase in which the head adjective is not final. Three other examples of this are:
1. We need a box bigger than that."
2. That is fruit ripe for the picking.
3. Which man is my dad? My dad is the man happy for his wife.