Gravity is viewed as a force because it is a force.
A force $F$ is something that makes objects of mass $m$ accelerate according to $F=ma$. The Moon or the ISS orbiting the Earth or a falling apple are accelerated by a particular force that is linked to the existence of the Earth and we have reserved the technical term "gravity" for it for 3+ centuries.
Einstein explained this gravitational force, $F=GMm/r^2$, as a consequence of the curved spacetime around the massive objects. But it's still true that:
Gravity is an interaction mediated by a field and the field also has an associated particle, exactly like the electromagnetic field.
The field that communicates gravity is the metric tensor field $g_{\mu\nu}(x,y,z,t)$. It also defines/disturbs the relationships for distances and geometry in the spacetime but this additional "pretty" interpretation doesn't matter. It's a field in the very same sense as the electric vector $\vec E(x,y,z,t)$ is a field. The metric tensor has a higher number of components but that's just a technical difference.
Much like the electromagnetic fields may support wave-like solutions, the electromagnetic waves, the metric tensor allows wave-like solutions, the gravitational waves. According to quantum theory, the energy carried by frequency $f$ waves isn't continuous. The energy of electromagnetic waves is carried in units, photons, of energy $E=hf$. The energy of gravitational waves is carried in the units, gravitons, that have energy $E=hf$. This relationship $E=hf$ is completely universal.
In fact, not only "beams" of waves may be interpreted in terms of these particles. Even static situations with a force in between may be explained by the action of these particles – photons and gravitons – but they must be virtual, not real, photons and gravitons. Again, the situations of electromagnetism and gravity are totally analogous.
You ask whether the spacetime is the force field. To some extent Yes, but it is more accurate to say that the spacetime geometry, the metric tensor, is the field.
Concerning your last question, indeed, one may describe the free motion of a probe in the gravitational field by saying that the probe follows the straightest possible trajectories. But where these straightest trajectories lead – and, for example, whether they are periodic in space (orbits) – depends on what the gravitational field (spacetime geometry) actually is. So instead of thinking about the trajectories as "straight lines" (which is not good as a universal attitude because the spacetime itself isn't "flat" i.e. made of mutually orthogonal straight uniform grids), it's more appropriate to think about the trajectories in a coordinate space and they're not straight in general. They're curved and the degree of curvature of these trajectories depends on the metric tensor – the spacetime geometry – the gravitational force field.
To summarize, gravity is a fundamental interaction just like the other three. The only differences between gravity and the other three forces are an additional "pretty" interpretation of the gravitational force field and some technicalities such as the higher spin of the messenger particle and non-renormalizability of the effective theory describing this particle.
While it's common to describe gravity as a fictitious force we should be cautious about the use of the adjective fictitious as this is a technical term meaning the gravitational force is not fundamental but is the result of an underlying property. The force itself most certainly exists as anyone who has been sat on by an elephant can attest.
There is a sense in which all forces are fictitious since they are all the emergent long range behaviour of quantum fields, so gravity is not unique in this respect. For more on this see Can all fundamental forces be fictitious forces?
Anyhow, the object responsible for the gravitational force is a tensor field called the metric, and when we quantise gravity we are quantising the metric not the force. The graviton then emerges as the excitation of the quantum field that describes the metric. As with other quantum fields we can have real gravitons that are the building blocks of gravitational waves and virtual gravitons used in scattering calculations.
Finally, you ask why it's necessary to quantise gravity, and this turns out to be a complicated question and one that ignites much debate about what it means to quantise gravity. However the question has already been thoroughly discussed in Is the quantization of gravity necessary for a quantum theory of gravity? While it's not directly related I can also recommend A list of inconveniences between quantum mechanics and (general) relativity? as interesting reading.
The principle reason that we want to quantise gravity is because Einstein's equation relates the curvature to the matter/energy distribution and the matter/energy is quantised. Einstein's equation tells us:
$$ \mathbf G = 8 \pi \mathbf T $$
where $\mathbf G$ is the Einstein tensor that describes the spacetime curvature while $\mathbf T$ is the stress-energy tensor that describes the matter/energy distribution. The problem is that $\mathbf T$ could describe matter that is in a superposition of states or an entangled state, and that implies that the curvature must also be in a superposition of states or entangled. And this is only possible if the spacetime curvature is described by a quantum theory, or some theory whose low energy limit is quantum mechanics.
Best Answer
An addendum to the answers of Daniel Grumiller and sb1:
The major difference of the gravitational field and other fields is that according to general relativity the gravitational field defines space and time and therefore defines the relation of events. It is true that it is possible to do an "arbitrary" split of a certain linear approximation of the gravitational field into a "flat background" and "waves" propagating on this background. In principle this kind of reasoning is a violation of the very idea that the gravitational field defines the background of spacetime in a holistic way, and it was the subject of a lot of discussions if this approximation is of any use.
This is considered to be settled by the observational evidence that bistar systems loose energy in exact the way that the "graviational wave approximation" predicts, as cited by Daniel Grumiller.
The existence of gravitons is a conjecture based on the assumption that gravitational waves exhibit the same quantum nature as classical waves, e.g. waves in classical electromagnetism. At the basis of this conjecture is the idea that it should be possible to split the gravitational field in a part defining the background, and then having gravitational waves propagating on this background and exhibiting the same wave-particle duality as other waves. It would then be possible to treat quantum gravitational effects in a semi-classical approximation.
Since there is no observational evidence for this, this conjecture is still the subject of controversy.
Some have a strong believe in the existence of gravitons, some think that quantum gravity needs a bigger conceptual step than only gravitions, and some do both, so, yes, there are different communities believing different things.
It explains everything in a classical setting with not too strong forces alright, but does not exlpain quantum effects or what happens when forces get so strong that singularities occur.
It is a force, it is an apparent force in the sense that classical GR says that you feel it because you live in an accelerating reference frame. Both statements are valid in the classical setting and are independent of the quantum nature of gravity, and in particular of the existence of gravitons.
Yes, see above (geometric model = classical setting, exchange particle = semi-classical approximation).
The gravitational field is fundemantally different from other fields (see above), and this has no connection to extra dimensions.
The problem is that if you are able to ask this question, you're already beyond the popular science level. I'd really like to recommend to you an introducory class on QFT and one on GR, there you'd get the best answer to your question :-)