Ah, good question. The radian is actually a "fake unit." What I mean by that is that the radian is defined as the ratio of distance around a circle (arclength) to the radius of a circle - in other words, it's a ratio of one distance to another distance. For an angle of one radian specifically, the arclength $s$ is equal to the radius $r$, so you get
$$1\text{ rad} = \frac{s}{r} = \frac{r}{r} = 1$$
The units of distance (meters or whatever) cancel out, and it turns out that "radian" is just a fancy name for 1!
Incidentally, this also implies that "degree" is just a fancy name for the number $\frac{\pi}{180}$, and "rotation" is just a fancy name for the number $2\pi$.
This actually addresses the edit to your question. Suppose that you had some object oscillating at $\omega = \pi/4\frac{\mathrm{rad}}{\mathrm{s}} = 0.785\frac{\mathrm{rad}}{\mathrm{s}}$, and you wanted to evaluate its position after 10 seconds. To get the cosine term, you would plug the numbers in, getting
$$\cos\bigl(0.785\tfrac{\mathrm{rad}}{\mathrm{s}}\times 10\mathrm{s}\bigr) = \cos(7.85\text{ rad}) = \cos(7.85)$$
and then you would go to a trig table in radians (or your calculator in radian mode) and look up 7.85.
However, suppose that you were measuring $\omega_0$ in degrees per second instead of radians per second. You would instead have
$$\cos(45^\circ/\mathrm{s}\times 10\mathrm{s}) = \cos(450^\circ)$$
If you go look this up in a trig table given in degrees, you will get the same answer as $\cos(7.85)$. Why? Well, remember that the unit "degree" is just code for $\pi/180$, so this is actually equal to
$$\cos\bigl(450\times\tfrac{\pi}{180}\bigr)$$
And $450\times\frac{\pi}{180} = 7.85$, which is just $450^\circ$ converted to radians. So now you have the same value in the cosine, $\cos(7.85)$. Trig tables listed in degrees already have this extra factor of $\frac{\pi}{180}$ built into them as a convenience for you; basically, if you look up any number $\theta$ in a table that uses degrees, what you get is actually the cosine (or sine, or whatever) of $\theta\times\frac{\pi}{180}$.
The answers are no and no. Being dimensionless or having the same dimension is a necessary condition for quantities to be "compatible", it is not a sufficient one. What one is trying to avoid is called category error. There is analogous situation in computer programming: one wishes to avoid putting values of some data type into places reserved for a different data type. But while having the same dimension is certainly required for values to belong to the same "data type", there is no reason why they can not be demarcated by many other categories in addition to that.
Newton meter is a unit of both torque and energy, and joules per kelvin of both entropy and heat capacity, but adding them is typically problematic. The same goes for adding proverbial apples and oranges measured in "dimensionless units" of counting numbers. Actually, the last example shows that the demarcation of categories depends on a context, if one only cares about apples and oranges as objects it might be ok to add them. Dimension is so prominent in physics because it is rarely meaningful to mix quantities of different dimensions, and there is a nice calculus (dimensional analysis) for keeping track of it. But it also makes sense to introduce additional categories to demarcate values of quantities like torque and energy, even if there may not be as nice a calculus for them.
As your own examples show it also makes sense to treat radians differently depending on context: take their category ("dimension") viz. steradians or counting numbers into account when deciding about addition, but disregard it when it comes to substitution into transcendental functions. Hertz is typically used to measure wave frequency, but because cycles and radians are officially dimensionless it shares dimension with the unit of angular velocity, radian per second, radians also make the only difference between amperes for electric current and ampere-turns for magnetomotive force. Similarly, dimensionless steradians are the only difference between lumen and candela, while luminous intensity and flux are often distinguished. So in those contexts it might also make sense to treat radians and steradians as "dimensional".
In fact, radians and steradians were in a class of their own as "supplementary units" of SI until 1995. That year the International Bureau on Weights and Measures (BIPM) decided that "ambiguous status of the supplementary units compromises the internal coherence of the SI", and reclassified them as "dimensionless derived units, the names and symbols of which may, but need not, be used in expressions for other SI derived units, as is convenient", thus eliminating the class of supplementary units. The desire to maintain a general rule that arguments of transcendental functions must be dimensionless might have played a role, but this shows that dimensional status is to a degree decided by convention rather than by fact. In the same vein, ampere was introduced as a new base unit into MKS system only in 1901, and incorporated into SI even later. As the name suggests, MKS originally made do with just meters, kilograms, and seconds as base units, this required fractional powers of meters and kilograms in the derived units of electric current however.
As @dmckee pointed out energy and torque can be distinguished as scalars and pseudo-scalars, meaning that under the orientation reversing transformations like reflections, the former keep their value while the latter switch sign. This brings up another categorization of quantities that plays a big role in physics, by transformation rules under coordinate changes. Among vectors there are "true" vectors (like velocity), covectors (like momentum), and pseudo-vectors (like angular momentum), in fact all tensor quantities are categorized by representations of orthogonal (in relativity Lorentz) group. This also comes with a nice calculus describing how tensor types combine under various operations (dot product, tensor product, wedge product, contractions, etc.). One reason for rewriting Maxwell's electrodynamics in terms of differential forms is to keep track of them. This becomes important when say the background metric is not Euclidean, because the identification of vectors and covectors depends on it. Different tensor types tend to have different dimensions anyway, but there are exceptions and the categorizations are clearly independent.
But even tensor type may not be enough. Before Joule's measurements of the mechanical equivalent of heat in 1840s the quantity of heat (measured in calories) and mechanical energy (measured in derived units) had two different dimensions. But even today one may wish to keep them in separate categories when studying a system where mechanical and thermal energy are approximately separately conserved, the same applies to Einstein's mass energy. This means that categorical boundaries are not set in stone, they may be erected or taken down both for practical expediency or due to a physical discovery.
Many historical peculiarities in the choice and development of units and unit systems are described in Klein's book The Science of Measurement.
Best Answer
Well I am going to come in here. I have read (and written) a lot around this subject and have pondered it for a long time. I think I can comment both on what I find makes sense, and also on what are the currents of thinking in the physics community.
This is a good question and you are right to ask it. There is no consensus on the answer! Some people think that space and time are so alike that it is just bizarre to have different units for them, as it would be to suggest we should measure distance north-south in, say, inches, and distance east-west in metres. So they would say that once we have understood this then the value is $c=1$ and it is dimensionless.
Other people prefer to say that time and space are not quite identical, because, for example, a timelike line cannot be confused for a spacelike line, nor turned into one by Lorentz transformation. Conservation laws, such as conservation of energy and momentum, make a highly insightful appeal to the difference between timelike and spacelike directions. So these people would say that it is useful to adopt units in which the speed of light evaluates to 1 distance-unit per time-unit, and it is not a dimensionless quantity.
The reason that there is not complete consensus is that there is no objective principle which asserts which is the correct or better choice. This is because the use of units in physics is always somewhat subject to human convention. When we say, of a mass for example, "it is 12 kg" then ultimately we are saying that the mass is 12 times larger than some other mass (one we have to define another way, or just resort to pointing to a physical object). I think the concept of physical dimensions is tremendously useful in analysis and in physical understanding, so I would not want to set it aside. For this reason I prefer to say that $c$ (and velocities more generally) are dimensional not dimensionless quantities. If we take light-seconds as our unit of distance, and seconds as our unit of time, then the value of $c$ is one distance-unit per time-unit.
Note that although time and space are parts of a single whole called spacetime, it is not quite right to say they are the same as one another. The future and past lightcone for any event is an objective idea, which separates spacetime into regions timelike-separated from that event, and another region spacelike-separated from it (and the boundary between them which is null). But nor are time and space completely distinct as they are in Newtonian physics.
In might interest you to know that in general relativity the mathematical framework adopted at the outset handles issues like this automatically as one "turns the crank" of tensor analysis, and as a result it is quite common to use sets of coordinates which do not all have the same physical dimensions. One could use $t,r,\theta,\phi$ for example. The use of the metric ensures that factors of $c$ are taken care of.
Further remark on units
After an exchange which you can see in the comments, I decided to add another remark. As I have said, the use of units in physics is to some extent a matter of human convention, but clearly some people judge that this does not extend to this issue in relativity, and they want to argue that only one formulation is correct. Here are the two formulations side by side. In the left column is the point of view where $c$ (and speed more generally) is taken as dimensionless. This can be done by reducing the number of physical dimensions employed to do physics. In the right column is the point of view where speed is dimensional.
(ls = light-second, s = second, m = metre) $$ \begin{array}{ll} c=1 & c = 1\; {\rm ls/s} \\ 3 \times 10^8 \,{\rm m} = 1 \,{\rm s} & 3 \times 10^8 \,{\rm m} = 1 \, {\rm ls} \\ 3 \times 10^8 \,{\rm m/s} = 1 & 3 \times 10^8 \,{\rm m/s} = 1 \,{\rm ls/s} \\ E^2 = m^2 + p^2 & E^2 = m^2 + p^2 \mbox{ and see * below} \end{array} $$ (*) In the case of an expression like this, if one thinks of $c$ as dimensional then it might be said to be an abuse of notation to miss out the physical dimensions when substituting the value "$c=1$ speed-unit" into any formula, but this is merely a matter of convention on what the formula means. In either case it has been taken as understood that when evaluating any given expression, units will be adopted in which $c$ has the numerical value of 1.