You can easily find a turn-key property in the $250k range with cap rates around 6.

Assets identified in this study include:Appliances: Oven/range, dishwasher, refrigerator, washer/dryer, microwaveElectrical and communication systems: TV connections and dedicated equipment outletsCabinetry and CountertopsDecorative light fixtures: Sconces, pendant fixtures, chandeliersInterior Finishes: Crown molding, wood panel wall covering, window treatments8.20% of the total depreciable basis was classified as 15-year class life.

I also have a world class designer on my team that helps get these kind of properties from the average $60-$75k/yr range to over $100k/yr range (for 3bd homes; more the bigger you go).  

So now you’ve gotten a wide variety of PERSONAL opinions, all based on the individual bias of the posters and their past relationship, feelings, fears, and experiences with THEIR debts.  

Start visiting building sites so you can meet the trades and see finishouts in different price ranges. 

- What are the best neighborhoods for buy & hold in this price range?

Certain areas in Hayward and San Leandro could meet most of your criteria but they are in the $1M+ range.

In Kansas City you can find properties with cap rates ranging from 5 - 7+Big difference in projected cash flows, and the most desirable neighborhoods are trading closer to that 5 cap rate. 

Once you know your price range and what you want to put down if using financing you can typically have a bank approval same day.Once approved if not paying all cash you talk with your agent or even self-explore and use the listing agent in most cases.

There are still plenty of neighborhoods where you buy small multi-family in that price range.