I have a birch tree in front of my house, more to the South East.

One third of this tree seems to be dead and some big branches are falling.

Shall I cut it and remove the roots completely, shall I cut just the dead parts, shall I keep the stump hoping it will revive?

Shall I plant another tree immediately after removing it? 

Any help will be greatly appreciated.

Plants are considered the secret weapon in Feng Shui. 

This is because, a home with specific landscaping and plants can or may attract birds.
Thus in this sense, a home that can attract birds imply that if it is nice place.

Other than that Feng Shui does not micro-manage your question.