I find a mattock is a useful tool for levering up paving slabs. I can't tell how large the slabs are from your photo, but be aware that the larger concrete slabs can be really heavy - if they're too heavy to lift and carry by yourself, then it's usually possible to lift them onto one edge and 'walk' them on their corners.
The wall can be broken up by attacking the mortar, either with a bolster chisel and lump hammer or SDS drill with a chisel attachment. Start with the top layer and work your way down. You'll need a skip for all the rubble.
It might be best to keep the wall bordering the communal walkway, so you might want to keep the pillar at the pavement end too, in which case take care when removing bricks near that pillar. When knocking boundary walls down, make sure first that they're your's to knock down ! It's always worth talking to the neighbours before your start, regardless.
It might not be wise to dig too much away near the house for fear of affecting the foundations, so I wonder if a slope (sloping from pavement level to the existing ground level next to the house wall) would be the answer ? If in doubt about the foundations, seek advice on this.
I'm not sure if planning permission is required for jobs like this. You'll also have to contact the council about lowering the paving stones along the edge of the pavement and that won't be cheap.
Lastly - I'm not sure your garden is long enough to get my car on there - are you happy that's it's long enough for a family car, to make it worth the effort ?
