solar system - Compared with similar stars, does the sun itself have properties that make life on Earth possible?

Other than being kind of in the "Goldilocks" range, meaning, not too big, not too small, (not to be confused with Goldilocks-zone, that's something else), there's nothing special about our sun. There might be something a little unusual about Jupiter's movement over the last 4 billion years and perhaps, something unusual in the just right formation of our moon that stabilizes the Earth's wobble some, but there's nothing special about the sun beyond being a good size.

Now, as far as the "center" vs "outside" of the galaxy, I've heard that too but I'm not sure how well established that theory is. The "OK" range might be fairly broad. Certainly you wouldn't want to be too close to a large star that goes nova, and you "might" not want to be on the outer edge of the galaxy assuming the galaxy has a kind of protective shield like the sun's heliosphere, though I'm not sure how important that is and I'm not sure anyone knows. It's possible there's an ideal ring within the milkyway, but I'm not sure anyone knows that for sure or knows how large it is.

A star can't be too large because there's probably a minimum lifetime for an Earth like planet to form. With a ratio of roughly the power of 2.5 a star twice the mass of our sun would only have a lifespan of 1.5 billion years before it goes red-giant. If you figure it takes 0.5-1.0 billion years for a planet to cool down enough to be a good place for life and for the early bombardment periods to complete and time for the planet to cool and form a magnetic field, 1.5 billion years is pretty tight to allow time for oxygen formation and evolution, so about 2 solar masses is close to the maximum ceiling for earth like planet candidate and that ignores the sun going through increasing luminosity as it approaches it's red giant stage. Ideally you might not want to go too much over 1.5 solar masses (about 4 billion years of main sequence).

Now, you could probably go a fair bit smaller than our sun, cause that would extend the lifespan significantly, but if you get too small, "Flare stars, coronal mass ejections and tidal locking could become issues. Still, there's a pretty good range in the Orange-Yellow dwarf/main sequence range and perhaps upper red dwarf where Earth like planets might be possible.

The bigger question (I think) is stable orbits and planet formation, I suspect. If a star has a gas giant that moves inward and becomes a hot jupiter or a gas giant with a very high eccentricity, I don't think a solar-system like that could maintain and earth like planet. Binary stars might have a hard time providing a billion or more years of stability. Similarly, if there are no large gas giants, There might not be enough early bombardment and water carrying comets, or, there might be too much bombardment later on, which isn't good either.

Other factors like a rotating core, a magnetic field, plate tectonics, and the right mix of gases, for example, too large a planet probably retains hydrogen and you can't get an oxygen atmosphere if there's too much hydrogen, so there's likely a maximum mass beyond which earth like planets couldn't be. There's other issues like snowball earth. Runaway Greenhouse. Planets really need a nearly perfect situation to be earth-like. I suspect there's not much room for error. Fortunately with a billion or so stars of the right size in the milky way, there's likely many potentially Earth like planets.

So, that's my "guess" is that earth like planets has much more to do with solar-system formation than anything else. The star simply has to be within a range of mass, maybe 0.1 or 0.2 to 1.5 or so solar masses. There's still so much we don't know about other solar systems that we really can't say how common earth like planets are. That's one reason I'm looking forward to James Webb Telescope. That should give us much better information on exo-planets and other solar-systems.