Depends on the priorities of the buyer. What's most important to them?
Having said that, I typically start with: (1) Price, (2) Number of bedrooms and baths, and (3) General location.
I don't start with school district. The way school districts are drawn, and how they change from year to year, that's just asking for trouble. If someone is absolutely insistent on living in a particular school district, I'd do a search of houses as close to the school district boundaries as possible. But I'd point out to the buyers that school district boundaries change. Schools get closed. Magnet schools can distort the picture, as well. A whole lot can change. Besides--speaking from personal experience--I can tell you that some schools with an excellent reputation are lousy (for certain students) and others with apparent drawbacks (how about a secondary school with 4,000 kids?) can be an excellent choice.
As for townships (or our equivalent here): No, not unless that's a very high priority for the buyer. Example: There's a small town--Falls Church--in Northern Virginia. It's considered very desirable and the schools are highly rated. But the houses cost more than houses just a few blocks outside the town. And the schools outside the town are still excellent. If someone said, "I absolutely need a house in the town of Falls Church," then--sure--that's what I'd search for. But I'd probably do another search with the same parameters except I'd look maybe a mile outside the town itself.
Where I am--and I know it'll vary based on location--buyers usually are interested in price, home size (bedrooms and baths), distance from where they work (downtown D.C., Tyson's Corner, Rockville, Fair Lakes), access to public transportation (in some cases), and type of property (single-family home, townhouse).
Hope that helps.