#1 is not so simple - most phones (Android at least) know how big their battery is supposed to be and will shut themselves off if they think they've consumed the whole thing. You usually need to root them and mess around with some undocumented firmware settings. (It's also a pain in the ass to crack open a phone to get access to the battery.)

I doubt "any 4V power source" is true AT ALL!

I once tried powering an android tablet with straight DC, turns it is EXTREMELY sensitive to slow power supplies, ie. it required a very beefy one. I had to add capacitors even to get it to work, it was really weird. But considering a battery can dish out some serious amps....

Also 4V sources are not common.

Lost in translation, I think. I mean you reuse the connector and controller and sensor from the original battery. Then you can connect any voltage 3.7-4.2 Volts to instead of Li-On cells.

There is of course the issue that a phone uses momentarily high current when transmitting. So there needs to be beefy capacitor too.

I have done this many times, of course. Especially on Ebooks, for which you cannot find replacement superthin batteries. You use original connector and any small lithiums in parallel. Discarded e-cigarettes are good source for small lithiums.