Hi again,
I looked over the inverting stage again and confirmed my thoughts of the other night. The 2.5V pseudo ground you show on your schematic (Vcc/2) has the same label for two very different points indicating a common point and the cause of the problem you experienced; one on the non-inverting input and the other in the feedback loop. These points are not common to the op amp operation if of high impedance (Z), and should not be mingled in that case. Below I'll provide one simulation with your design with one amp showing the commonality of the source with high Z, another with separate high Z sources and a third with a single low Z source and the graphic results for of all three. The maximum input for that configuration with that gain before clipping is about an impractical 35uV, the input level used in the sim; it may work in a sim but probably not in the real world.
Look closely at the top two plots. Note that the signal at PG1, which should not be in the feedback loop at all, is 180 deg out of phase with the output. I'll let you think about that implication and determine the impact on the output and why V(outa) is degenerated to almost nil.
Also, when I asked for the input drive level above I wanted to know what the AC input voltage to the first amplifier to bring into focus the full scope of the issues you were experiencing. Working backwards, I have given you the input limitations for the two different configurations shown below.
The schematics you attached above showed a total gain of over 5800 when coupled together and in an operational configuration, yet you say in your post #2 you expected a gain of 1000. I have included a third simulation of the circuit with the objective, for which I think you are shooting...gain=1000. Also, note with a single-ended supply of only 5V the maximum input level before clipping is ~2mv peak or 1.41mv rms.
One last thing...I used the MC33078 in the sims as that is what I had in my library. The LM833 in your schematics is a clone of the MC33078 in all major aspects of performance so it's apples and apples in that respect.
I feel like I have left something out/hanging, so if you have other questions or need clarification of my comments, please ask so they may be cleared up.
Good luck with your project,
Merv