This is more a set of comments than an answer per se, but too long to fit in a comment.
Signal from oscillators: 8Vpp 1~20KHz with an offset of ~10V with a new 9V battery.
So the issue is how to couple this to another stage which can amplify it, but at the same time set an appropriate input voltage DC offset suitable to the next stage.
The obvious solution would be to use any amplifier design with reasonably high input impedance, and AC couple to it via a capacitor, so for example a cap from OSC1 to R8.
"The main problem is, on Q1 base, where the signals meet, there's no signal." Whatever voltage signal is at Q1 base will be quite small because the impedance at Q1 base will be small compared to the 1 Meg input resistors. (Especially for frequencies above the knee of the R5-C7 highpass filter.)
So the voltages at Q1 base may well be only 1/100 or 1/1000 of the signals into R8 and R9. In any case what you are more concerned with is the AC currents through R8 and R9 (and thence into Q1-base).
And probably also of concern is the DC voltage at Q1-base -- is it in a sensible range to bias Q1 to operate in it's active range, say with 3 to 4 V DC at Q1 collector? Since you have a 100k collector resistor on Q1, that suggests you are expecting a DC Ic of around 0.03mA to 0.04mA, and thus a DC voltage of rather precisely 0.03V-0.04V across R5 (and not, for example, 0.08V), but there's nothing to set a suitable voltage on Q1-base to make that happen so far as I can see.
Finally, what is the role of C9, 10nF? In parallel with R11 that appears to create a filter that will attenuate output above 160Hz or so, working to considerably suppress the signals in your range of interest, 1 kHz-20kHz.
It's difficult to say anything about what you wrote after "My mission: be able to make its output signal usable" because you don't show a schematic of what your did and it's hard to guess.
FWIW, if you feed an AC audio signal via a capacitor into a voltage follower (which has a high impedance input, hence shouldn't disrupt the source of the signal), you are going to get an output voltage that follows the input voltage. That's assuming you've set the DC level at the follower input to something reasonable. There's not much that can go wrong there, so we need to see exactly what you did that might have cause this to fail.
Bottom line, it looks like your challenge here may be simply understanding how amplifiers work (either op amps or with discrete transistors) and how to satisfy their input requirements for signal voltage or current, impedance, and DC bias (aka offset). Perhaps reading up on that topic might allow you to navigate more satisfactorily?
To summarize, you want to know what opamp characteristics you need to buffer a reference voltage.
The two most obvious are low offset voltage and unit-gain stability. Any offset voltage is a error directly added to the output. Without unity-gain stability, the thing will oscillate in voltage follower (unity gain) mode.
Other parameters depend on more specific circumstances. For example, will this thing be subjected to wide temperature swings. If so, low offset across a wide temperature range is important, not just offset at 25 °C or whatever.
Other issues come up depending on what supply voltages you have available relative to the reference voltage you want to buffer, and how stable those supplies are.
Do you expect the load to have a significant capacitive component? If so, you have to look at stability more carefully.
In any case, it would be a good idea to filter both supply inputs with a ferrite chip inductor or two followed by caps to ground. That way you don't have to count as much on the active power supply rejection capability at high frequencies.
Best Answer
The emitter follower can only source current, it cannot sink it*. The LM6132 has an input bias current of (typically) 110nA at room temperature. That current flows out of the op-amp input.
If you connect it to a 500K load to ground, you'll get around 50mV of output offset. To make this go away you could add an E-B resistor- something like 1K should work very well.
*If you had a -10V supply it would probably appear to work since the transistor E-B junction would break down. This is not a good thing. Even with your +/-5V supplies it would be a good idea to add a reverse-biased diode such as a 1N4148 across the E-B junction (in parallel with the resistor I mentioned above).