Is paganism compatible with Christianity?

You can take compatible in a number of directions and the answers, accordingly, varies:

A. Compatible meaning theologically interchangeable - answer is no, clearly.

B. Compatible means holding similar doctrines - answer is no.

C. Compatible means holding similar values - answer is USUALLY yes. Just look at the values espoused in Buddhism and Hinduism, not to mention Judaism and Islam. I'd say very similar in terms of the Golden rule, kindness, purity, loyalty, respect for authority, etc.

There are other directions we could go in as well, and the answer would change.

It depends which angle you're looking at it from. Is Paganism compatible with Christianity? is, to be frank, a hopelessly vague question.

Paganism is a very broad category. Most Pagans would accept Jesus as a deity. A few might include him in the their personal Pantheon. Most wouldn't. It is certainly possible, though, for a Pagan to do reverence to Jesus within a Pagan context. From that perspective, yes, they are compatible.

Christianity is also a broad category. Many Christian traditions are unashamedly borrowed from Pagan sources. There are aspects of Easter which are drawn from a spring fertility festival, and Christmas and Halowe'en also show signs of Pagan influence. Various customs and rituals (such as churches being oriented toward the sunrise) may also have a Pagan origin. Does that mean that Paganism is "compatible with" Christianity? Well, it certainly means that some traditions are compatible.

Really, the problem here is that your question is very poorly defined. There's no good clue as to what you're actually asking. I've answered two questions. I suspect that neither is the question you're actually asking.

Let's move on. It's not just traditions and rituals which are derived from Pagan sources. Some Christian beliefs are, too. As an example, the Gospel writers Matthew and Luke developed a story that Jesus was born of a virgin, with accompanying signs and wonders. Why did they tell this tale? Because it was, at the time, part of the standard mythos of a hero. Mithras was born of a virgin. Julius Caesar was born of a virgin. In fact, here's a list I nabbed from somewhere:

Julius Caesar, Augustus, Aristomenes, Alexander the Great, Plato, Cyrus, the elder Scipio, some of the Egyptian Pharaohs, the Buddha, Hermes, Mithra, Attis-Adonis, Hercules, Cybele, Demeter, Leo, and Vulcan.

Actual Christian beliefs taken from Pagan sources are, of course, a more contentious matter, since many Christians will disagree about the origins of these stories. Looking at religion at the outside, as a series of memes borrowed and reshuffled, this seems pretty clear. We borrow beliefs from the broader society in which we find ourselves. That's how these things work. It's how we all work. We could not be otherwise and yet be human. The Trinity doctrine, for example, is frankly weird, and there's vanishingly little evidence that it was part of Christianity in the first century. And yet now the vast majority of Christians accept it and call it a defining part of their faith. Again, it was borrowed.

Does that make Christianity "compatible" with Paganism? Well, the word compatible remains hopelessly ill-defined in this context, so I can't say. I suspect not. A mere convergence of belief in certain specific areas, however those beliefs are derived, is not the same as "compatibility", by most definitions.

Let me put that another way. Christians have a religious framework which prioritises belief over practice, which presents a hierarchical view of the universe, and which is history oriented. Pagans, by and large, have a religious framework which prioritises practice over belief*, which has a far less hierarchical view of the universe, and which is more cyclically oriented. They are very very different. If in certain specific areas, they have a couple of beliefs or traditions in common, they are no less different overall. I don't know what definition of compatible you're using, but by most definitions, Paganism and Christianity are incompatible.

* Pagans with very different beliefs can participate in the same rituals; Pagans with very similar beliefs may have incompatible rituals.

Per your own quoted dictionary, no. By definition they are incompatible.

Pegan: ... observing a polytheistic religion ... who is not a Christian ... irreligious. (dictionary.com definitions #1, #2 and #3)

Per the every major creed in Christian history, no. The whole purpose of these creedal statements was to show how Christianity's beliefs are set apart from any other and define the God we believe in and how men are to be saved.

We believe in one God, the Father Almighty, Maker of heaven and earth, and of all things visible and invisible. (Nicene Creed)

Per the Ten Commandments, no. God defines the rules for his chosen people as being incompatible with any others.

You shall have no other Gods before me. (Exodus 20:3)

Per Jesus NT teachings, no. By his own statement, Jesus is incompatible with all other belief systems.

I am the way, and the truth, and the life. No one comes to the Father except through me. (John 14:6)

In summary. No.

Quite simply, there might be similarities because some aspects of true religion and even divine truth are seen in the wider culture, but at their core they are fundamentally incompatible.

Christians understand some of the similarities to be echos of general revelation that people instinctively understand some things about God and others to be perversions of the truth. We see in the garden when the serpent talks to Eve that he loves to take nuggets of truth to bait a hook of lies and we see this pattern repeated throughout history.

Likewise not all of Christianity is pure and unadulterated; in the same way that many pagan practices are perversions of truth, many non-Biblical customs and ideas have been allowed into the "culture" of Christianity. Some may be harmless or simple tarnishes dimming its vibrance, others poison sapping its life and dissuading people from the truth.

No. Jesus said "I am the way, the truth, and the light. No one comes to the Father but by me."

It's an exclusive statement.

As is "Thou shalt have no other gods before me" and "Thou shalt not make any graven image" (which I take to mean a "god" we create with our hands or with our imagination.)

The entire history of the Jewish people, as recorded in the old testament was one of being saved (first from Egyptian slavery, then by various captivities and judgments throughout history), turning toward God, then after a generation or so, turning away from God to Pagan gods. The entire history recorded in the Bible would indicate that paganism is not compatible with Christianity or Judaism, from which Christianity came.

However, there are certain teachings that the two might have in common. Certain things are undeniably good. Pagan religions may promote selflessness, kindness, mercy, and a whole host of other teachings that are compatible with Christian teachings. In my answer, I'm not addressing whether there are commonalities, as this is a different question from "Are the two compatible".