In the long term, quality matters more than quantity too IMO ... even if you have great cash flow on paper but the property value and rents are going down over time when inflation adjusted, then it will likely be difficult to collect that cash flow in those type markets in the short term while you are losing purchasing power over the long haul ... that business model is more like leasing a used car than it is REI in my opinion, which is not to say that you can't make money but it is a vastly different and less passive business model. 

2-3% cap rate means the prices are over inflated and WHEN the market turns it will be the first place to get hit hard.at 2-3% this this probably running a negative cash-flow.

Odds of winning the lotto?

Good plan IMO except for a few things ... 1)Don't buy turnkey ... buy something where you can put in some sweat equity and force some appreciation; this will give you extra experience and also enhance your returns to scale faster/easier. 2)Do not ignore appreciation rates; cash flow day one is great, but you want to make sure you buy in a neighborhood that will hold its value and has steady, reliable rents after inflation at the least and at best increases in value; if not, then your cash flow could just be your initial investment getting trickled back to you in monthly installments while you are losing purchasing power on the property. 3)Keep your possessions down to a minimum in a smaller space; this is good financial advice anyway but doubly important if you will be moving every few years.

You should hire your own pool service so you don't have to pay the inflated cost with the PM nor trust the tenant with pool care.

But, then again, I'm in an odd market.If there's no utility costs factored in, that means each unit should have it's own water heater, own air heater, own A/C (if it has them) ... that's a lot of potential capex.

Not that there is a magic number, but I do believe your odds of success at both learning the business and making a profit are better when buying more than just one note (in 2nds world).

Odds are that they would accept your offer and award you the contract to perform the work required. 

We think the house would look odd.

This doe not seem to be an odd situation at all in this market condition.