SMS-Code does not appear on MacBookPro screen, although iPhone says so. I tried already several things that are posted in the net, especially taking my devices of my apple account. Nothing worked :-). Any other suggestions?

If the problem is that cellular text messages are not forwarded to or from your iPhone, please follow the instructions in this support article, under the heading

To use Continuity for SMS and MMS with your iPhone and your Mac, iPad or iPod touch

Make sure that at least one email address is enabled on the phone for use with iMessage; otherwise the activation code won't appear.

CAPSEL wrote:

When "enables" means start the conversation with an e-mail. ... This could be it. My conversation starts with my phone number.

No that's not it, all my SMS messaging originates from my phone number (it has to), even from my Mac and iPad.

You do have to have an email address enabled as a messaging point of contact on all devices, but you don't have to actually use it. I suspect that it is the email address entry that links non-SMS-capable devices to your iPhone, so make sure you're using the same address (sounds like you are) and make sure all devices are on the same Wi-Fi network when you attempt to link them.

First, make sure that your devices meet the requirements (see SMS section at the bottom):

System requirements for Continuity on iPhone, iPad, iPod touch, and Mac - Apple Support

If you're good with all that, then try signing out of iMessage and FaceTime on all devices, maybe even throw in a Reset of iOS devices, then sign back in (iPhone first, then others) making sure that all devices are on the same Wi-Fi network to establish the initial connection.

What versions of iOS and OS X are you running?