Can a planet be tectonically inactive and still retain a magnetosphere and protected atmosphere? How does that work?
How else could a planet retain a thick atmosphere like Earth's for extensive periods of time?
Note, I'm not asking on Earth science SE as, well, the question is not about Earth. I was hoping there was some sort of astronomical theory for other potential habitable planets.
You make the connection that a lack of plate tectonics on a planet indicates a "dead" core and thus said planet has no magnetosphere. As such, I'm going to interpret your question as, can a planet without a magnetosphere retain an atmosphere long-term? As proof, I offer up Venus.
Venus' Magnetic Field
Venus is a planet without a magnetosphere generated by a core. It is thought that the cause for this is Venus' slow rotation rate (of nearly 243 days) and a lack of convection, allowing for bulk motion in the core. As I'm sure you know, you need moving charge to make magnetic fields and Venus' core just isn't moving. As such, we see that Venus is a tectonically dead planet - it's surface is roughly 500 million years old, whereas Earth's surface gets recycled every 100 million years or less due to our plate tectonics.
Now, Venus is not entirely devoid of a magnetic field. Ironically enough, it's lack of a magnetosphere allows for the generation of a magnetic field by it's atmosphere. Because the Sun's radiation is more or less directly hitting the atmosphere, Venus has a strong ionosphere. When you get lots of charged particles moving around in an atmosphere, you get a magnetic field. But on the whole, this field is very, very weak compared to a true magnetosphere such as we have on Earth.
I found this source which talks a lot about this concept and why Venus doesn't have a magnetosphere. Check it out to get a lot more in depth detail.
Venus' Atmosphere
So, Venus has no appreciable magnetosphere (or plate tectonics). Why does it have an atmosphere? And boy does it have an atmosphere. The surface pressure on Venus is estimated to be $\sim93\:\mathrm{atm}$.
In short, the answer is that the inundation of solar wind against an atmosphere is not necessarily the major contributing factor to atmospheric loss. It can be, but not always. For example, Mercury, another planet with a weak (but non-zero) magnetosphere, has no atmosphere (if it ever did) because it is so close to the Sun that the solar wind likely blew away that atmosphere long ago. Venus on the other hand is far enough away that the solar wind just can't strip the atmosphere. Here I'm going to quote wikipedia directly (emphasis mine).
A lack of magnetic field does not determine the fate of a planet's atmosphere. Venus, for instance, has no powerful magnetic field. Its close proximity to the Sun also increases the speed and number of particles, and would presumably cause the atmosphere to be stripped almost entirely, much like that of Mars. Despite this, the atmosphere of Venus is two orders of magnitudes denser than Earth's. Recent models indicate that stripping by solar wind accounts for less than 1/3 of total non-thermal loss processes.
Atmospheric Loss
If the solar wind isn't the contributing factor in atmospheric loss, what is? The answer to that is a process known as Jean's Escape. To put it simply, for gas particles in the atmosphere to escape into space, they need enough energy to climb out of the planet's gravity well. Some particles will have that energy and thus escape into space. Over time, the atmosphere bleeds off, little by little (this is happening for Earth too!).
The factors that contribute to the rate at which a planet loses its atmosphere are such things as the the planet's mass and radius, and the mass of the atmospheric particles. Let's look at Venus. It is comparable in mass and size to Earth and so has a reasonably appreciable gravity well. For something to escape Venus it must be traveling at $10.4\:\mathrm{km/s}$ (compared to the Earth's $11.2\:\mathrm{km/s}$). But, for Venus at least, the important factor there is that the atoms and molecules in its atmosphere are heavy. It is almost entirely ($\sim97\%$) carbon dioxide which has a mass of $\sim44\:\mathrm{amu}$. That means, the chances of such a massive particle getting the energy to escape is pretty small.
Out-gassing
Just one more point to add to this. One may argue that possibly the atmosphere is/can be replenished continuously, but that won't work here because we're assuming the planet is tectonically dead. You can't really have out-gassing on a planet with no active surface.
Conclusion
There are many factors that determine atmospheric escape. Different planets will lose their atmospherics for different reasons. However, it is entirely possible for a planet, under the right conditions, to maintain an atmosphere long-term, despite lacking a global magnetosphere. As we can see from Venus, the conditions are generally that the planet should be sufficiently far from the star, its atmosphere should be sufficiently dense and comprised of heavy particles, and the planet itself should be large enough to have an appreciable gravity well. If these conditions are all met, a planet may retain an atmosphere without having a magnetosphere to protect it.
