First Sentence

Let us be a bit nitpicky. Your first proposition is OK, but it just means you are doing it for her. That may be with or without her consent. As a non native English speaker, when I hear "on behalf of somebody", I tend to think of a designated representative or helper. So, if my wife had instructed me to call, I'd probably go with:

Ich rufe Im Namen meiner Frau an.

The second proposition is also correct, but it means you are calling because of your wife.

Second sentence

There is a small flaw. You should say:

Es geht um meinen Sohn.

Third sentence

Not bad, but watch the articles! It should be

Ich möchte um einen Termin mit Doktor Krueger bitten.

or

Ich möchte um einen Termin mit dem Doktor/Arzt bitten.

In these cases "mit" implies a meeting with the doctor himself, not his assistant. I typically use the more abstract:

Ich möchte um einen Termin bei Doktor Krueger bitten.

Sidenotes

Typically, the use of your first name is too personal when calling an unknown.

I personally never use "ich moechte gern" in requests, but that may be a personal preference.

First Sentence

Let us be a bit nitpicky. Your first proposition is OK, but it just means you are doing it for her. That may be with or without her consent. As a non native English speaker, when I hear "on behalf of somebody", I tend to think of a designated representative or helper. So, if my wife had instructed me to call, I'd probably go with:

Ich rufe im Namen meiner Frau an.

The second proposition is also correct, but it means you are calling because of your wife.

Second sentence

There is a small flaw. You should say:

Es geht um meinen Sohn.

Third sentence

Not bad, but watch the articles! It should be

Ich möchte um einen Termin mit Doktor Krueger bitten.

or

Ich möchte um einen Termin mit dem Doktor/Arzt bitten.

In these cases "mit" implies a meeting with the doctor himself, not his assistant. I typically use the more abstract:

Ich möchte um einen Termin bei Doktor Krueger bitten.

Sidenotes

Typically, the use of your first name is too personal when calling an unknown.

I personally never use "ich moechte gern" in requests, but that may be a personal preference.