I need help to sketch the vector field F(x,y) = 2xi - yj by choosing four relevant vectors in each quadrant. Then I need to find and sketch the equation of the field line that passes through the point (x,y) = (2,2). It is possible to choose vectors where x = 0 and y = 0. Could you please help?

For each x,y the point (x, y) is the point where a vector starts and F(x, y) is a pair of its components. For x=y=0 we have F(x,y)=vec0 and we cannot draw it with an arrow because it has no direction.
For the given point (2,2) its field vector is F(2,2) = (4, -2), it is the direction vector of the corresponding line. This means the slope of this line is y/x = -2/4 = -1/2 and the equation is  y-2=-1/2 (x-2), or y=-1/2 x+3.
To draw relevant vectors, choose some integer arguments of F(x,y). It is simple to start a vector F(x,y) at the point (x,y): we just need add (x,y) and (2x, -y) and obtain (3x, 0). This way all vectors point to the x-axis.
The attached picture uses the starting points (1,1), (1,2), (1,3), (1,4) for the first quadrant, (-1,1), (-2,2), (-3,3), (-4,4) for the second, (-1,-1), (-2,-1), (-1,-2), (-2,-2) for the third and (1,-1), (2,-1), (3,-1), (4,-1) for the fourth. Actually, the field is symmetric across x and y axes.
http://tutorial.math.lamar.edu/Classes/CalcIII/VectorFields.aspx