Single-supply op-amp behavior with 0V input
You are basically right, except for the choice of 741 to use over the 0-5 V range.

The 741 requires some headroom against both supplies.  For 0-5 V operation, a CMOS "rail to rail" opamp is a much better choice.  Once of the Microchip MCPxxxx series is likely a good fit.

The circuit you show (other than, again, poor choice of opamp) is valid and is common enough to have its own name of "voltage follower".  The voltage gain is basically 1, but the output impedance can be orders of magnitude lower than the input, making this a useful trick in some circumstances.

As to why the simulator doesn't act like a real part, it's not a real part.  Apparently that restriction on opamp behavior was not coded into that simulator.  Simulators are only as good as the people who wrote them and the users that drive them.  Go hook up a real opamp and gain some real world intuition about how these things behave.