Do force fields physically exist? Or are they just a region of space where the forces are acting around its source, for example a magnet? But if they are just regions, and not physical objects, then how can the Earth's magnetic field be reshaped to a "teardrop" by the solar wind?
 A: A force field by definition is a vector field, i.e. a region of space where to each point a vector (=force) is assigned. As such it is not a physical object, however you can of course probe/measure the field with a suitable test object (mass in case of gravitational field, charge in case of electrical field...).
The solar wind, being a plasma, carries the sun's magnetic field to the earth (interplanetary magnetic field). So at the earth you will have two contributions to the magnetic field, the magnetic dipole from earth (which is more or less symmetric) and the field from the solar wind. A test probe would be affected by the sum of these two fields.
Now, due to the Lorentz force, particles from the solar wind get deflected and travel around the earth. This leads to similar effects as in hydrodynamic when you have an object in a stream of fluid (see Magnetohydrodynamics), making the total field asymmetric (and different/reshaped from the magnetic dipole field).
A: Physics uses fields, a mathematical concept, both at the classical framework and at the the quantum mechanical framework, to model observations and predict outcomes of experiments. The answer of user1583209 is within the classical framework for the use of 'fields' as you are asking about the magnetic field.

A field can be classified as a scalar field, a vector field, a spinor field or a tensor field according to whether the represented physical quantity is a scalar, a vector, a spinor or a tensor, respectively. A field has a unique tensorial character in every point where it is defined: i.e. a field cannot be a scalar field somewhere and a vector field somewhere else. For example, the Newtonian gravitational field is a vector field: specifying its value at a point in spacetime requires three numbers, the components of the gravitational field vector at that point. Moreover, within each category (scalar, vector, tensor), a field can be either a classical field or a quantum field, depending on whether it is characterized by numbers or quantum operators respectively. 

Italics mine.
As far as classical fields go, they are a mathematical construct dependent in most cases  centered on sources and their kinematic behavior. An exception is the electromagnetic wave  which propagates independent of the source, but due its its motion it cannot be considered as "regions in space"
The mathematical theory which successfully describes quantum mechanical effects in  the microcosm of atoms and particles is quantum field theory. These are fields that are quantum operators operating on a quantum ground state. All particles in the standard model of particle physics are assigned a quantum field , extending over all space , and is assigned a ground state on which a creation operator for the field, the electron for example,operating on the ground state will create an electron, and an annihilation operator will destroy it. The ground state is zero, if there are no particles created/annihilated so in this sense also the concept is mathematical to allow for calculating the behavior of interacting particles.
For people interested on how classical fields emerge from quantum mechanical fields this link explains it  but it needs mathematical tools.
A: Fields are genuine physical objects, which carry energy, momentum, and sometimes charge (as in Yang-Mills theory and General Relativity). Fields can exist without any sources, as, for example, electromagnetic waves or hypothetical glueballs.
A: My gut feeling is that the question originates from a mental model of a world full of physical bodies, i.e. things. A drawing in this world depicts a physical object: A chair, a table, a house. Planets. I can manipulate the smaller ones: Take them elsewhere, turn them over. The bigger ones influence each other: The sun attracts the planets which orbit around it. 
I can only draw things which exist, in this palpable sense. 
Now somebody draws a magnetic field. I interpret your question in this sense: Does this drawing depict something that really, physically, bodily exists? The answer, unfortunately, goes in the opposite direction of what your question implies. The physical objects we see are actually systems of fields. The chair is a collection of nuclei and electrons held in place by electrostatic and -dynamic interaction; and on a closer look we cannot maintain the "bodily existence" of any of these "particles" in any "solid-matter" sense. They are waves, quantum states, probabilities, processes, which in the particular case of the chair I'm sitting in right now congeal to an overwhelmingly probable interaction with my behind resting on it. Phew.
The answer therefore is that a magnetic field like the earth's is exactly as real as anything else around you; not more, and not less. But the reason is that the air of "reality" our daily surroundings emanate is rather deceiving.
