Part of this will get into a debate over what the "first universities" were. As an Arabist I'm more sympathetic to the idea that institutions like the Nizamiyyas were a form of university than many European medievalists might be. At the very least, without getting into the definition of a "university" these madrassas were clearly a form of higher education.

In the Islamic world in the middle ages, who was eligible for higher education and what their benefits were were closely intertwined. While for much of the Middle Ages the political classes of the Islamic world consisted of either steppe peoples or slave soldiers, education offered a path for non-military figures to join the ranks of the elite by becoming members of the Ulema, the religious class of scholars within Islam.

Despite their clearest societal role being as religious leaders and scholars, because so much of late medieval history in the Islamic world the territories of Islamdom were dominated by external steppe peoples, the Ulema actually played a very powerful political role in mediating between the rulers and the ruled. Despite the constant flux of dynasties (Buyids, Ghaznavids, Seljuks, Ghurids, etc. etc.) because the Ulema were able to preserve the legal norms of Islam, they preserved a certain measure of societal continuity despite the total political chaos.

Access to this powerful political class was, at least theoretically, meritocratic, and even people from humble origins could rise to great heights as a religious scholar. Although of course the requisites of having access to education was inherently favorable to men from cities, and, of course, women had no role whatsoever.

This relationship between the foreign ruling class and the local intermediaries has been termed the Amir (Warrior Prince)/Ayan (Local Notable) system.

In terms of "second jobs," I can't speak to knowledge of any of scholars who took up such a job, but I can say that even a junior scholar's preparation would have qualified them for other jobs. Entry to a madrasa would by definition and at a bare minimum have required the student to have memorized the Quran. Qualification as a religious judge would have further required memorization of a set number of hadiths and their respective Isnads (the chains of transmission from the Prophet to the recorder of the hadith) (depending on the time period, the number could reach the thousands.) Having memorized the Quran would allow such students to become Quran reciters, which was a paying position, as well as teachers of the Quran. Although being certified as a Qadi (religious judge) was a far more prestigious (and politically official) position than either of these.

So to give a bit of TL:DR for the Islamic world's answer to your questions, in order:

Who was eligible? Everyone, but with a bias towards urban dwellers.

What were the benefits? Access to a powerful political class that was otherwise unavailable to non-military figures.

Would they have gained a higher salary than the alternative? Yes.

Major or specialization? The curriculum was by definition religious, but depending on the era in question there were different schools of law to choose from.

Second jobs? I don't know of any scholars who describe having taken such jobs, but even entry-level students would have been qualified to perform a number of roles that were paid.

Sources: Marshall Hodgson's The Venture of Islam goes into quite a lot of detail about how this political system worked in Islamdom in the Middle Ages with detail about social institutions including Madrasas.