It is quite straightforward. The built in mesh is structured, but if you use mshr to generate the mesh, it becomes unstructured. See minimal example below.
from dolfin import *
from mshr import *
domain = Rectangle(Point(0,0), Point(2,2))
mesh = generate_mesh(domain, 10)
File("mesh.pvd") << mesh
No, In dolfin, in general there is very little support for quadrilaterals cells. In dolfinx you can use quadrilateral elements, with built in meshes, or unstructured meshes creates with for instance gmsh.