How to authorize iPad running on iOS 7.0.3.?

You can NOT get that dialog on an iPad running iOS 7.

On an iPad running iOS you need to enter the password immediately followed by the code (BOTH in the SAME password dialob box at the same time).

Do it in the order that I described in my previous reply.

FIRST -- get a verification code : see: Get a verification code and sign in with two-factor authentication - Apple Support

SECOND -- do what you need to do to trigger the following:

Third: Enter both the password and the code in the above dialog box.

You type the code into the same box where you type the password -- i.e., the password and the authorization code are entered back to back into the password dialog. If the password dialog is already gone, first get an authorization code, start over to trigger a new password dialog and then enter both.

See the following for other ways to get the code: Get a verification code and sign in with two-factor authentication - Apple Support

From your original post: "I try to download app from the App Store: I enter my Apple password, authorize on another device, but a dialog box for entering the verification code on iPad doesn't appear..."

1. Get a code as described in my previous reply.
2. Do whatever you need to do (e.g., start completely over trying to download an app) to get back to the part of your original post that I have underlined and changed to bold above.
3. Enter both the password and the code in the password box.

See the following from Two-factor authentication for Apple ID - Apple Support

If you use two-factor authentication with devices running older OS versions—like an Apple TV (2nd or 3rd generation)—you might be asked to add your six-digit verification code to the end of your password when signing in. Get your verification code from a trusted device running iOS 9 and later or OS X El Capitan and later, or have it sent to your trusted phone number. Then type your password followed by the six-digit verification code directly into the password field.

I wouldn't say you were dumb -- but I was starting to feel like I might be in not coming up with a better way to explain it!