Is there some specific time period/date where we say, "before x human endeavor is considered archaeology?"
Is there debate?
Is this question totally off-base because they're totally separate fields?
In the US, archaeology is treated as a subfield of anthroplogy. As such, a standard definition is "the study of past people through physical remains. " Written accounts technically aren't archaeological so the division is rather simple in the Americas; you don't have any pre-Colombian historians. Timewise, you don't see history before writing.
But what about afterwards? What if I'm excavating a shipwreck from the War of 1812? It certainly fits the definition of archaeology, bur we have plenty of historical documentation about the USA military at the time. We call this "historical archaeology." Is it history or archaeology? Yes.
Then what about Classical Archaeology, or Chinese, or Egyptian, or Mesopotamia? We have written accounts stretching very far back for them, but they must be heavily supplemented with physical remains. You'll find that in Eurasia, many archaeology departments are under the history one. So there really is no point in time where one begins and the other ends.
Rather, there are shifts in one's utility and focus over the other. Historical documents like photos, journals, newspapers, and news broadcasts can tell us a lot more about the Vietnam War than an archaeological study of battle sites or student rally sites. If I'm studying the political career of Roman Emperor Vespasian, I'll get more from documents than from any artifact. But if it's a Roman settlement in far north Britain, there might be a few mentions of the town name in a decree or two, but an excavation can tell us all kinds of things.
Archaeology deals with people, thier lives and identities, cultures and their interactions, and such. This compliments history, rather than being apart from it.
While archaeology is its own discipline, of course (well, sometimes), I like to see it more as a method of study than anything. It's not so much when something stops being history and becomes archaeology. In fact, there is a type of archaeology known as historical archaeology, and this is essentially doing archaeology with the aid of documents and archives (i.e. primary sources for historians). In North America, historical archaeology would be the archaeology of the post-contact period. I've excavated sites that were no more than 150 years old, and that's definitely archaeology, but we can also figure out the names of the people who lived at that farmhouse or whatever because there are historical documents. Some archaeologists excavate even more recent things, and some archaeologists work with living people (an area known as ethnoarchaeology) because seeing how someone makes pottery, for instance, can help us understand how it was done in the past.
To really simplify, written documents are the main source for historians, and beyond a certain point (about 5,500 years ago at the absolute earliest) there was no form of writing anywhere in the world, and so archaeology is the only way to learn anything about people who lived before writing developed in any given part of the world. But even when writing and record keeping was around it only discussed a small segment of society so archaeology is still very useful.
But it is more complicated than that, even, because material culture (i.e. artifacts, ecofacts, and features: the tools, buildings, ornamental objects, foods, etc. that we make and use) is extremely central to what it means to be human, and archaeology is the only social science discipline that really focuses on material culture.
And that brings me around to another point: at least in North America, archaeology is not history whatsoever; it is indeed a totally separate field. Archaeology within North America is typically considered to be a sub-discipline of anthropology. I am an anthropologists who uses archaeology. In the rest of the world archaeology does tend to be more closely related to history than it is in North America, but even then it is often its own discipline that spun out from history at some point, and often it has ties to geology too when you look at the development of the field (some archaeology came out of studying early human evolution, where fossils that were 50,000 - 1,000,000 years old were often found in quarries).
But it gets more complicated yet. Classics departments usually have archaeologists who primarily study Greece, Rome, and the eastern Mediterranean region. They are classical archaeologists, just as much an archaeologist but even though there are some at my university, I don't know them or have any interaction with them. They are a different discipline altogether. There are also archaeologists who are art historians.
So, the best answer I can give you is that it depends. I wouldn't say there's debate about whether something should be considered archaeology or history and it really does not matter what we call it. But the answer is yes to your other two questions. In certain settings. Most of the time. Kind of. And like we archaeologists always say, context is everything.
I study early medieval Europe (a time period where very few written sources survive), so I rely heavily on archaeological evidence and usually find myself with a foot in each discipline. I've found that while history and archaeology are very complimentary approaches to studying the past, the two disciplines often have different methods and questions for getting at the past.
Archaeologists tend to think more like scientists than most historians do. Archaeology usually involves manipulating a lot of raw data, and you're more likely to see archaeologists using statistics, computerized mapping (GIS), and very explicit hypothesis testing. Many archaeologists work in a lab, and their studies depend heavily on forensic analysis of bones, chemical analysis of pottery, carbon dating, etc. Archaeologists are often concerned with demonstrating the validity or limitations of these laboratory methods in the course of their research, so archaeological studies often discuss not only the data, but also how effective the methods were in analyzing that data, and whether the approaches could be improved in future research.
Historians do many of the same things (including using statistics, computer modeling, etc), but generally focus more on the results of their analysis than the methods themselves. It's not that historians don't like to talk about methods (they do) - but history tends to emphasize storytelling more than archaeology. While archaeologists usually reconstruct cultural systems, demographic trends, etc, historians try to trace sequences of events. And as a result, historians tend to be less interested in discussing method for its own value in their articles and monographs.
So when an archaeologist approaches a set of data from material culture, the archaeologist is more likely to use that data as a case study to show how a particular method of analysis can tell us interesting things about the past. In the process, we'll also learn about past culture systems. Historians are much more likely to use that information to frame a narrative within its wider social and material context; but the events are the main focus, and method is only discussed to the extent that it's necessary to justify the historian's particular interpretation of the evidence.
This answer is, of course, criminally generalized, and I can think of several dozen examples off the top of my head that contradict the pattern I'm describing. And its further complicated by historical archaeologists studying archaeology in periods from which texts survive, and historians (like myself) studying history in periods where most of the evidence is archaeological. But in the most general terms, the difference boils down to social scientific (archaeology) vs. literary / storytelling (history) approaches.
hi! you may be interested in this earlier post
and a couple AMAs that expand on archaeology