Everything regarding Starlink's plans would be speculation at this point, but we can look at existing comparable LEO satellite-to-mobile services, i.e. Globalstar and Iridium.
There, the answers are:
> What frequencies does this operate at?
1-2 GHz. These are also common terrestrial mobile communication frequencies.
> How well matched are the antennas in most phones?
That I don't know for Iridium (I think they're at least somewhat specialized), but for Globalstar, Apple is able to pull it off with their newer iPhones.
> How much more power are you using to be able to do this?
Iridium uses around 1-2 W – that's significantly more than LTE (I believe most devices only use about 125 mW, but I wasn't able to find precise numbers), but about the same as GSM/2G, which current phones largely still support (although I'm not sure whether they use that much transmit power practically still).
There, the answers are:
> What frequencies does this operate at?
1-2 GHz. These are also common terrestrial mobile communication frequencies.
> How well matched are the antennas in most phones?
That I don't know for Iridium (I think they're at least somewhat specialized), but for Globalstar, Apple is able to pull it off with their newer iPhones.
> How much more power are you using to be able to do this?
Iridium uses around 1-2 W – that's significantly more than LTE (I believe most devices only use about 125 mW, but I wasn't able to find precise numbers), but about the same as GSM/2G, which current phones largely still support (although I'm not sure whether they use that much transmit power practically still).