I believe Musk is, like Teledesic, planning to use the Ka-band. That means the smallest antenna would be the size of a large laptop (and that's only if phased arrays become cheap enough). So, it would work for a car, plane, boat, or house, but would be too big to use while walking.

How powerful would a terrestrial satellite amp need to be in order to be effective? how cheap would it need to be to sell?

But one could imagine a city where many shops have satellite internet and open WiFi, so that in effect you can walk around with a phone and use this.

Will it still need gyro mechanism?

There are satellite (I think Ku band) dishes for larger (~50ft) boats nowadays. They are using gyroscope assisted motors that wouldn't suffice for compensating rolling motion of a smaller boat.

The laptop-sized antenna he mentioned doesn't need a gyro because it uses phased-array tech. Essentially you resolve signals in any direction (up to a resolution given by number of elements and array size) by carefully combining them with certain phase offsets such that signals from all but one direction cancel out. This adjustment is made on the fly to track a signal with changing direction.

I see.

Would something like pCell would be an improvement here?

pCell is just phased array technology.

Directionality and sectioning has been used since the beginning of cell technology: first with cell splitting and then with cell sectoring, and recently more aggressive splitting (smaller cells) -- it's an ongoing evolution. The kind of tracking this has required is very easy to do: once your signal is stronger in a competing sector for a significant amount of time, switch sectors. The same principle can be applied to phased arrays but the small number of sectors makes it much more tractable computationally and in terms of hardware.

A phased array in a city environment needs to track in real time the position of users, with enough feedback from them to make sure your prediction matches with their position well enough, and use this information to continuously adjust sub-nanosecond (depends on the freq/bandwidth) phase offsets to array elements. In principle it`s even possible to do multi-tower coherent communication. Needless to say, this tech is expensive right now, and I imagine hard to get to work reliably.

MIMO is also closely related, the difference being traditional MIMO actually adjusts amplitudes only (non-coherent) of multiple data streams to create channels (after a linear transformation at the receiver). This means the signals are still ominidirectional for your wi-fi MIMO. Being non-coherent means it's much cheaper.

Not if I'm wearing my urban sombrero!