I'm not convinced its needed but if we are going to build it start in Scotland or the North. At the moment all it feels it will achieve is drawing more people to London and the South East as Birmingham becomes a suburb of London for commuting purposes.
Investing outside of the South East might encourage people and companies to move elsewhere which can only be good for house prices (in that hopefully there would be levelling effect and less demand on the SE).
But the capacity problems are not as severe in the North. You would not be able to use any extra capacity up there for through trains from the south.
Possibly not but it is very underinvested in and slow. If you develop the North and Scotland you may not need all those trains from the South. In my mind improving travel into and around London is only going to encourage more people to work there. Bit like the roads argument.