Could a planet be at the center of the universe, yet still have a sun?

Not as commonly thought of, no.

One of the things that we learn in orbital mechanics is that the planet orbits its star, not the other way around. More generally, the less massive body orbits the more massive body. When the difference in mass is large ($m_1 \gg m_2$), that's a good enough approximation, and for simplicity we can even consider the masses to be point masses; when the masses are of similar magnitude, it gets a little more complicated and the barycenter – the common center of mass – ends up somewhere between the centers of mass of the two bodies. When the situation is sufficiently extreme, even the formulas we take for granted break down entirely and completely different equations are needed to describe reality. Compare also the Newtonian and relativistic orbital velocity formulas (formulas 6 and 20, respectively) (Internet Archive link in case the page changes).

The only way to make a star orbit its planet is to make the planet significantly more massive than the star.

That, however, presents obvious problems, not the least of which is that by the time you go beyond 13 Jupiter masses, you get a brown dwarf instead. You would need a planet of, at the very least, very near that mass if you want it to be considered to orbit the star. What you would effectively have, then, is something very close to a system of two brown dwarf stars, one of which failed to gain the mass necessary to start its fusion processes.

On larger scales, it's not practical at all. There is no scientifically plausible way to explain why the Milky Way, with conservative estimates of its mass being 7e11 solar masses ($7 \times 10^{11} M_\odot $), would orbit the Earth, which measures about 3e-6 solar masses ($3 \times 10^{-6} M_\odot $). This difference is 17 orders of magnitude; said another way, given that the Earth has a mass of about 6e24 kg and a large satellite might have a mass of maybe 1e4 kg (10,000 kg), making the Milky Way orbit the Earth would be on a similar scale as making the Earth orbit a human-built satellite.

While in principle you can, as suggested by Separatrix, define the frame of reference such that the Earth is at the center, if you do so then you end up trying to explain relationships that eventually led to Kepler's laws of planetary motion and even later to modern orbital mechanics. Picking an Earth-centered frame of reference for your calculations will help with objects that are gravitiationally bound to Earth (which is why we often do it with Earth satellites etc.), you will have a much harder time using such a model to explain, for example, the movement of Saturn's moons. As a model will need to be a good fit for all available data, these headaches are likely to cause more grief than they are worth.

And of course, from a strict point of view, there is no such thing as "the center of the universe". For some elaboration on that, see for example What is in the center of the universe? on the Astronomy SE, or Does the universe have a center? on Physics SE, as well as the material linked from those questions and their answers.

posted over 8 years ago

Canina‭

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