You can plant under it if you take care to choose varieties that can stand the shade under the tree and it will be very dry under there too, depending on what type of tree you plant.

Yes, its the plants that would suffer rather than the tree. Also choose the tree carefully as some get far too big for the average garden.

Refer to RHS Encyclopaedia for final size of tree. The
roots will be the same size, so might cause trouble in
years to come under the brickwork of your house.

Wild flowers grow in poor soil under hedges (when they are allowed to grow by crop spraying farmers) would be worth
considering.