Why radian is dimensionless?
The fact that an angle is dimensionless is mostly a matter of convention. Indeed you can associate a dimension with an angle, and still remain consistent. Quite a few well known formulas need to be changed in that case though.
For example, you quote the arc length as $s = R\theta$. This is obviously no longer homogeneous in dimensions if $\theta$ is not dimensionless. However, if we remember that actually $s$ must be proportional to the angle $\theta$, and that $s=R$ for $\theta=1\,\mathrm{rad}$, we conclude that the true form of that relation must be $$s = R\frac{\theta}{1\,\mathrm{rad}}.$$ (When asuuming that $\mathrm{1\,rad = 1}$ is dimensionless, it yields the original expression.)
Another thing is that we can no longer feed an angle directly to analytic functions like sin, cos, exp, etc. Indeed, these are conveniently defined via a power series, e.g. $$\exp(x) = \sum_{k=0}^\infty\frac{x^k}{k!}.$$
If $x$ has a dimension, this expression makes not whole lot of sense. To be correct, we would have to write $\sin(\theta/\mathrm{rad})$ when talking about the sine of an angle.
As of my understanding, these are some of the reasons why one decided that an angle should best be left dimensionless. Especially since angles are relevant in mathematics, where -unlike in physics- one typically does not care about dimensionality of quantities.
What are the advantages of assigning a dimension to angles? Mostly, the additional dimensionality carries a lot more information. Consider frequency $f$ and angular velocity $\omega$. In the SI system both have the same dimension, namely 1/time. If angle had its own dimension, the unit of $\omega$ would be $\mathrm{rad/s}$! We could differentiate these quantities based on their units! Depending on how you introduce the new dimension, torque and work might also no longer share the same unit.
If you are interested, here is a very readable article that explains a possible way of introducing an additional dimension for angles.
A dimensionless quality is a measure without a physical dimension; a "pure" number without physical units.
However, such qualities may be measured in terms of "dimensionless units", which are usually defined as a ratio of physical constants, or properties, such that the dimensions cancel out. Thus the radian measure of angle as the ratio of arc length to radius length is one where the units of length cancel out.
Yep, you can work with dimensioned angles.
The formula for the arc length is
$$l=\aleph\theta,$$ and that for the area of a sector
$$a=\frac{\aleph\theta r^2}2.$$
The universal constant $\aleph$ is in per-angle-unit, and $\aleph=1 \text{ rad}^{-1}=0.0174533\cdots\text{ deg}^{-1}$.
For example, when considering an harmonic movement,
$$e=A\sin(\aleph\omega t)$$ where $\omega$ is in angle-unit $s^{-1}$ and $e$ in $m$.
$$v=\dot e=A\aleph\omega \cos(\aleph\omega t)$$ is in $ms^{-1}$.
Another universal constant worth to know:
$$\Pi=3.141593\cdots\text{ rad}=180\text{ deg}.$$
denotes the aperture angle of a half-circle.
It fufills
$$\aleph\Pi=\pi.$$
Hence the famous Euler formula,
$$e^{i\aleph\Pi}=-1.$$