I have a Eurocentric focus and I will be using examples from primarily from Europe. The concepts will be universally applicable however. There is a great deal of subtlety to the concepts, and giving definitive answers is hard.

How often would they have eaten meat?

Not as often or as much compared to most modern westerns. The amount of meat would have depended on the time of year. Feasts would have been a major event that would have been called for meat. In colder climates you would butcher all animals that you couldn't house for the winter, and this was often more than you might expect in the number of animals (Pig litters are around ten piglets that mature in 6 months time as an example. Meaning that a farmer could have reliable 6+ 250 lb Hogs each year.). Much of this meat would be preserved (Salted, smoked) to be used through out the year.
People living on bodies of water or near rivers and lakes would have eaten more fish as meat, where more inland people would have eaten more livestock.

Would all food have been cooked at home, or would bread and "processed foods" be sourced from outside?

This would depend on the staple food. In places with flat breads or where people ate unmilled grains mostly (Such as rice or quinoa) they would normally cook at home. When it comes yeasted foods such as bread, that could have been home or made by a baker. Often small villages might have community ovens that everyone could use.

Did habits change significantly between regions, a across the rural-urban divide?

Yes. This topic could be a book in of itself. Things would have varied based on the food native to a region. People in the Low Countries might have eaten endives (a type of lettuce), where in central Europe you might see more kohlrabi. Even in modern countries you see differences. An example would be France; where in the south people eat more olive oil and in the north more butter.
Differences can be big and small.

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Do you have any further questions?