The LM308 is not a rail-to-rail opamp; the 600 mV minimum input voltage you see is the voltage drop across the output transistor inside the opamp (the output on the negative side is actually a cascade(*) configuration). The same applies to the positive side.
The 3.8 Volt is probably because your operating voltage is too low and some internal transistor goes into saturation. Increase the power supply to 10 Volt and you should get better results.
(*) Can't come up with the proper term right now.
About the gain being stated as \$A_V=3\$, the complete relevant text is
R3 and R4 form an output voltage gain stage whose gain, \$A_V=3\$, is reduced to unity at high frequencies by C1 to maintain stability.
What this is saying is that R3 and R4 form a voltage divider so that
$$v_o = \frac{v_{out}}{3}$$
where \$v_{o}\$ is the voltage at the output of the op-amp IC.
Or, turned around,
$$v_{out} = 3 v_o.$$
This works because the negative feedback around the op-amp will cause it to push or pull current from its output pin to make it work.
The overall gain of the circuit is 33, as you calculated.
, it is stated that the gain of the whole circuit is 33 while the gain of the output is 3 . So you can have 2 gains in an op-amp ?
No, the "stage" formed by R3 and R4, with gain 3 doesn't really involve the op-amp.
But even within the op-amp integrated circuit itself, of course every stage in the design can have a different gain value.
So one is the gain of the input to the Vout? and the other is the gain of the op-amp output to the actual Vout?
33 is the gain from \$v_{in}\$ to \$v_{out}\$
3 is the gain from \$v_o\$ to \$v_{out}\$. I think Tim's comment does a better job explaining it than I could:
If the voltage on the op-amp output is not equal to 1/3 of the output voltage (ignoring C1), then current will flow in it's positive or negative power pin. That forms a negative feedback loop composed of the four external capacitors and the op-amps (internal) output stage.
Best Answer
This is a common characteristic of rail-to-rail input op-amps. There are really two front ends and there is a transition between them at some common-mode voltage. R-R output is irrelevant.
If you care about Vos to that extent, you can choose another type of op-amp. Maybe a chopper type, but they have other subtle (and not so subtle) imperfections.