I'm only abble to shed some light on the first part of your question, aside from the issue of networks and recommendations that other user already tackled.

Like today, Universities issued diplomas and kept records of them. These diplomas usually were handwriten or printed in paper of excellent quality, like vellum (calfskin), materials with great durability as we can see from documents with hundreds of years that are still in excellent shape. They would also bear a wax seal (like this) or one embossed in paper of better quality (like this one on the bottom left) when cheaper paper started being used for the printing. The seals were a way to attest their authenticity, as were the signatures from the dean and other competent authorities. Each University or even each Faculty would have their own seal and a metal matrix (I have this type of matrix in mind) to reproduce them, so to if someone were to try to counterfeit a diploma they would have to make a new matrix.

The wax seal, also used in other official documents since the Middle Ages (e.g.: this diploma from 1597, Blibliotéque National de France, or other great examples compiled by the University of Nottingham) , was kept in a small box, sometimes made of metal, attached to the document by a ribbon. Having had to inspect some diplomas in this format from the 19th century, I can attest that this was the most fragile part, since the ribbon could rip into the paper or catch on something when you take them out of the storage.

At least until the early 1800s, Universities, like Coimbra and Oxford and Louvain, issued diplomas in Latin, which a lot of scholars were familiar with at that point. In the 19th century you can already find documents in the vernacular language, although some institutions kept the Latin, but I'm unable of getting into when and why these changes occurred, although I'm certain this had to do with nationalist sentiments as well as the widespread of litteracy.

Because you asked about Europe, I looked around to be sure of somethings, and found this Diploma of the University of Oxford conferring upon Samuel Johnson, of Pembroke College, the degree of M.A.held by the British University. I'll reproduce this part Tof the description of the document since it is very illustrative for your question:

After 13 months, Johnson dropped out, probably due to lack of money. Later, he made several failed attempts to get himself an MA on the basis of his literary achievements – firstly from Oxford in 1738, and secondly from Trinity College, Dublin.

By the summer of 1754, Johnson had finished the onerous task of defining more than 40,000 words. He then travelled to Oxford to research the Dictionary’s prefatory sections. While there, he enlisted the help of two friends in his quest to get an MA ‒ the poet Thomas Warton (1728‒1790), and the librarian Francis Wise (1695‒1767). After a flurry of letters, the diploma was rushed through, just as the Dictionary was being printed. On 21 December, Johnson wrote to Warton that he would ‘keep back the title page’ so that he could insert his MA. Finally, on 25 February 1755, the diploma arrived in the post. It was just in time for ‘A.M.’ ‒ short for Artium Magister ‒ to appear next to Johnson’s name when the Dictionary was published in April.

So If you need to prove you received your degree, you would have this document and it couldn't get much better than this. But you could always write to your university and they would look into their records and send letter attesting that you indeed had received that degree, signed and sealed for good measure, and present that as proof as well.

To end this with an anecdote (I won't say the names because these are my supervisors' findings and weren't published yet), I was researching a foreign painter who applied for a position as a teacher in an Academy and therefore had to prove he had the qualifications needed. Unfortunatly he was unable to to do that because, as he says, he came to the country by boat and it capsized, so he lost every document and other proof of the prizes he had receiveid in his country. It could very much have been true, but he didn't get the job.