Humming and Transmission issues
Regarding your power supply. You have created a dual supply with two batteries. Batteries have internal resistance and so should be bypassed by capacitors. Any resistance in your return circuit can create ground loops. When a dual supply is created using batteries, there is a risk that it will go lopsided if one battery loses voltage faster than the other one.
You have a number of sloppy choices of part values in your circuit. For example at the bottom, you have a virtual ground mixer whose input impedance is is only 100 ohms. First of all, that is a low input impedance to ask the other op-amp to drive. Secondly, it loads down the 10K pot that you have in front of it. Weird things will happen.  When the pot is turned all the way up, the impedance is basically just the 150 ohms. The pot's impedance to ground doesn't matter because it is much larger. When you turn it halfway, the impedance goes up because now you have a 5K branch of the pot in series with the input impedance. If you want a potentiometer to be linear, it should have an impedance which is at least 10X lower than the input facing the wiper.  If you have approximately equal input (say 10K linear pot, 10K resistance) then it's nonlinear in a useful way because it approximates audio taper.
The gain on your upper op-amp, from the PC speaker, is less than unity, because you have a 1.5K feedback resistor and a 2.2K input resistor.  That low gain, combined with the terrible loading presented by the next stage (at least when the volume pot is turned all the way up) is probably why you have inadequate signal going back across the transformer.
The 450uF coupler between the transformer and your op-amp-ology is quite overblown
. If you achieve a decent input impedance in the mixer, you can use a much smaller cap. Since you have a dual supply, you should not need it at all. The only thing it does is protect the transformer winding from DC flowing across it, if your twp batteries become unbalanced (so that the output of the upper op-amp develops a serious DC offset at quiescence). Similarly, the coupler to the headphones should also not be needed except for the threat of unbalanced batteries. If the headphones are 600 ohms, what would be a good coupler value, if we want low end down to 20 Hz? (And do we?) For flat down to 20Hz into 600 ohms, the cap would have to be 130 uF. Yours is spec'd for elephant ears: flat down to 5 Hz.   You can bridge op-amps to create an amplifier that doesn't require DC coupling to a speaker. Be sure to use parts that can drive 600 ohm loads well, like NE5532 or LM4562. The noteworthy NJM4556 can drive 150 ohm loads. 
Regarding the hum; it is obvious from the sound files that the hum occurs when the phone is plugged in, whether or not the PC speaker is plugged in. It is 50 Hz hum. (I didn't notice you're European until examining this file! When I play it side by side with a 50 Hz tone generated in Audacity, the beat frequency between the two is very low: the frequencies coincide very closely.) There is a noticeable difference in the level of the hum when the PC audio is jacked in, suggestive of a ground loop between the PC and the phone system. (That ground loop would be external: in your device, you do not have a ground between the two but rather an isolating transformer.) But a more plausible hypothesis is this. Signal from the transformer secondary spills into the top op-amp circuit. You have two different connections from the transformer to that circuit. The PC audio output almost certainly has a low impedance to ground, so when you plug in, imagine that you're connecting jack's tips to the sleeve (if not with a conductor then at least with a capacitor for AC signals: the PC could have a coupling output capacitor that blocks DC; we don't know). This change in the circuit topology is likely the reason for the difference in the level of the hum.  About fixing it, no ideas. If I had to guess, I would say that problem (or at least a problem) is that the local telephone circuit from the box to your phone jack is an unbalanced, unshielded line: just two wires twisted together. That is susceptible to noise.