I have one official first name which appears on my ID card. I have another first name which friends call me by. Is it possible to use my unofficial first name and official family name to publish scientific papers? How can I prove to someone that I am one of the authors?
It depends on the field, and in particular on the venue in which you are publishing. It's possible that some journals and conferences might have policies about this, which would of course override anything you read here. But in general: nobody checks (or cares) whether the name you put on a published paper matches your official name. So if you use an informal variant of your name, e.g. if your name is "Stephen" but you publish as "Steve", nobody will bat an eyelid. If you use a nickname or publish under a pseudonym, it would be a little odd if the nickname is something that sounds very informal, but still, it probably won't cause problems.
What people do care about is building a record of your work, and putting a face to the name if they know your face. So you should (1) be consistent with the name you use to publish, and (2) make sure other people in the field know that it refers to you.
The answer of David Z covers all important aspects regarding the name choice. In order to prove your authorship to someone, you would have to show correspondence letters with the publisher. You will typically receive several of those, the most important one being the acceptance letter stating that the publisher will publish your manuscript. If you show these to anybody, it should be sufficient proof that you are an author, especially if the address on the letter can be clearly linked to you.
Note that this is not just an issue with using pseudonyms. Anybody with a sufficiently common name will face exactly the same problem.
I have a legal (official) transliteration of my surname (family name) from the Greek alphabet to the Latin one on all my official documents. I consistently use for many years now another transliteration (just one letter difference) of my surname in all the publications I have produced until now. I have never had a problem with the name tag in my conference badge or the hotel reservation or paying the conference registration or anything. Nobody seems to care until now. The one and only exception was just one time that I needed to issue a travel visa for entering a country; in that case the embassy staff questioned me why my official surname did not match the invitation letter sent by the conference organizers to invite me to present in the conference. The visa was issued without any hassle in the end.
It is permissable to adapt one's name for publication.
But to avoid the circumstance that you fear, i.e. that you may have difficulty "proving" your publications if your legal name is somewhat different, you should ensure that your nom de plume (assumed name for an author) is unique in your Department at the time of publication.
Your proof lies in your academic affiliation at the time of publication. Once you are the only person with that name (or assumed name) then you are safe and sound.