I cannot speak to the rest of Europe, but in the Gaelic, Welsh, and Norse cultures, your last name was dependent on the name of your father or your clan. If you were in Medieval Ireland, and Scotland, your name could be Padraig Mac Donnal, Mac in this instance being "Son of..." They would also use the designation "Ua" or "Ui" to indicate membership to a larger family like the Ui Nell or Ua Brian, and someone can correct me bit I believe they are gender specific and mean something along the line of grandson/daughter of, which is where we get the Angliscised O' in front of Irish names. In Wales, Mac was replaced with Ap, as in Llewlyn Ap Gruffudd, Gruffud being his father. In Norway, Sweden, and Denmark you would add "son" or "dattr" at the end of the name to show your relation to your father. So Harald Godwinson, would be Harald son of Godwin. Helga Bjornsdattr would be Helga, daughter to Bjorn.

After the Norman's invaded the British Isles, they started taking the census in the form of the Doomsday Book, and people needed a better way to be counted. There were undoubtedly thousands of John's and Michael's and Mary's, so they started using either geographical markers or trade markers for last names. This is how we get names like Burroughs, Ford, Hill, Woods, etc. We also get trade names like Smith, Forester, Cooper, Fletcher, Hunter etc. A quick look at the most common French surnames will show something similar to English names in that many are geographically or occupationally connected.

Surnames were used before the Doomsday Book (1086 AD), such as those names in Rome that were connected so closely with family fortune and reputation. But after Rome fell the practice fell out in Europe as well with resurgence of Gaelic and Germannic tribes that held different naming customs.

I am not certain of any first name restrictions, however I am sure the Church wanted to keep the names Christian which is why we have the Gaelic translations of so many names. For example Seamus comes from James, Sean from John, etc. In an effort to stamp out pagan practices, the Church looked down on naming kids non Christian names, which, was still common in Ireland, Scotland, and Wales for centuries, and still is today.

Since most nobles were named similarly after saints, it could be argued that there were no widespread restrictions on naming your son Edward when the local lord or king was also Edward, however I could be wrong in this respect.

There is a lot of generalization going on up there, some names have their own unique origins that do not have the same naming conventions, but it's a start.

I hope that answers at least part of your question.

Although another user might elaborate if there were sumptuary laws somewhere at some point regulating what name could and could not be given to newborns, in the meantime you might be interested in this answer of mine examining some aspects tied to surname transmission in pre-census Italy.

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