How was music written back then? How does it compare to what we see today? Are there any examples of what sheet music looked like back then compared to how we write sheet music today? When did sheet music start to look like the traditional sheet music we see today?
Here's a previous answer relevant to the evolution of musical notation.
Check this book. You can download it in pdf form. It includes lots of examples of different types of notation.
When did sheet music start to look like the traditional sheet music we see today?
Being pedantic, I'd say 19th century. 18th century music is perfectly readable, but you will start to find some different things many modern musicians will not be able to figure out at sight (ornaments, figured bass, white eights/quavers, dots to the left of a note that mean it most be played before it appears to be notated). 17th century music is still pretty readable in general, but tablatures were quite peculiar, plus the problems of 18th century notation and some more. 16th century music looks notably different but is still mostly readable (performing it usually requires some specialized knowledge).
The amount of required improvisation increases as you travel back in time.
Now, being able to read the notes doesn't mean you have the required knowledge to play that music. There are many issues with tempi, ornaments, articulations, phrasing, etc. Notes that look equal but are not supposed to be of the same duration...
The title is about music theory in general and I just focused on notation because of the other questions. A LOT of things happened between 1100 and 1600 in music theory. Way too many to try to address them in one answer. Please feel free to ask about other areas.
I'm not near any of my books right now so I'll source this later, but the 12th to the 16th century is too much musicology for this type of question. Could you narrow it down more or become more specific with your question please?
BROADLY speaking, written music started out around the 1200's with anonymous monks who used neumes to indicate whether a pitch was higher or lower than the previous one before it. However, this did not indicate specific pitches. Eventually, around the beginnings to the middle of Renaissance-era music, specific pitches were becoming indicated and that slowly evolved into what we have today. Reminder: this is only a broad overview
