Lots of questions.

Grounding

1. If you're struck by lightning, that's because there was such a strong potential difference in charge between you and the clouds that the air itself broke down, forming a plasma, and equalizing that potential difference in a very short amount of time. Since it's a lot of energy in such a short amount of time, that means a lot of power. I hope you brought a change of underwear.

Schematics

2. In schematics, symbols like ground and Vcc (power supply) are special in that multiple instances of them typically mean "connect these all together".
3. The LM386 is somewhat unique among ICs because its inputs have a resistor-to-ground built in. Imagine a 50k resistor from pin 3 to pin 4, connected inside the chip. There's one from pin 2 to ground also, but in this circuit that's moot because pin 2 is grounded. You are supposed to understand that by looking up and reading the datasheet, noting especially the parts about "input resistance" or if you're lucky, the "simplified schematic" which shows the inner workings of the chip.

Amplifiers

4. It's hard to guess why one amp didn't work and another did with so little information to go on. A concept that is probably going to help you is "impedance matching" which means basically that some amplifiers are designed to efficiently drive certain types of loads, and are less efficient with different types of loads. PC audio outputs are generically classified as "line level" which means they're optimized to deliver a couple volts into a 600 ohm load. BUT they will still likely "work" for headphones that are 30 ohms. Why? The impedance mismatch is not very noticeable. Instead of delivering a couple volts it delivers a couple millivolts. But headphones are efficient and so the resulting signal is good enough for many purposes.

5. The phone you linked to has a 2.5mm "headset jack", which probably means you can treat it like a headphone jack on a typical audio player, or even like your PC's audio output.

6. "boosted" output on the phone - don't worry about this right now. The phone has enough output to drive headphones, which means it will work with the circuit in your image. You don't need a "booster" and the person you're talking to either doesn't understand this or isn't framing it in a helpful way for you.

7. Push-Pull - way off topic. This is a kind of amplifier arrangement which is often used inside integrated circuit chips. In fact, there's at least one inside the LM386. But learning about this is tangential to today's goal of "make the phone louder".

8. I think this circuit will work with the Panasonic phone.

Capacitors

9. Capacity is measured in Farads, or often (because a Farad is huge) micro-Farads, abbreviated µF. Ohms are a measure of resistance. Very different things.

10. The function of the 0.05 µF capacitor is a complex subject. For now, just take for granted that it forms a "filter", together with the 10 ohm resistor, that prevents this particular circuit's amplifier from wasting power. If you omit these components or deviate from the suggested values too much, the amplifier will oscillate at a high frequency (too high for you to hear) which will waste power and may harm the speaker.

11. Yes, you can use a 220µF instead of the 250µF capacitor. The only difference will be slightly less bass response from the speaker. But if you're using this with voice it probably won't be missed.

Your first approach is unlikely to work very well because analog multiplexers don't exhibit good resistance matching between channels - in the case of HEF4067B, it's listed as up to 25 ohms mismatch. They also exhibit variable on resistance over voltage:

Your second approach should work because current won't be flowing through the MUX, meaning the mismatched resistance won't have much effect on the output reading. It may still result in offset errors that you need to correct per-gauge, but you'll have to calibrate them independently.

Another option would be to look for an analog frontend suitable to your application. There exist precision DACs with differential input, programmable gain, and multiplexed frontends, such as ADS1243 from TI, which would allow you to implement your entire front-end - mux, amplify and convert - in a single IC.