You can get a clearance through the LCY Class D but doing that flight in a single engine plane would not be too wise as you can't comply with the glide clear rule.
The story on this one has varied and it has been claimed that the CAA has ruled on it in the manner above. I don't have a reference though.
But yes it would be Thames Radar. For any Class D transit you call up the controlling authority, whose frequency is marked up on the VFR chart. For the Southampton piece you call up 120.225, or 119.475. These are on the chart.
A MATZ is not controlled airspace. It is "polite" to call up somebody - usually the nearest radar unit.