There is really just one underlying principle here: Any equation we write down should not depend on any arbitrary choices we made in order to define the quantities. All the examples you can discuss can be understood in this principle.
Can't add a vector and a scalar. Well, of course a vector is three numbers and a scalar is one number, so, for example, $\mathbf{v} + v$, , where $v$ is a speed, i.e. a scalar with units of velocity, doesn't even make mathematical sense. But we could add imagine adding one component of a vector to a scalar, i.e. $v_z + v$. But, this quantity shouldn't appear in a fundamental law of physics because our choice of what axis to call the $z$ axis is completely arbitrary, and if we made a different choice our equations would look different. But this is situation-dependent. For example, if we were discussing physics in a background uniform gravitational field, then we can use a convention where $z$ points along the direction of gravitational field. This is not arbitrary because the gravitational field sets a preferred direction. By declaring that we are going to call that particular direction the "z-direction'', it makes sense that any equations we then write down will only hold for that particular choice of $z$-axis. That is why the equation for the gravitational potential energy in a gravitational field, $U = mgz$, is valid even though $z$ is a component of the displacement vector $\textbf{r}$. However, you can still translate this equation into one that is valid for any choice of axes, namely $U = m \textbf{g} \cdot \textbf{r}$, where $\textbf{g}$ is the gravitational field vector.
Can't add numbers with different units. The point is that we normally work in units of physics which are chosen completely arbitrarily. If time $t$ is measured in seconds and position $x$ is measured in meters, then it
makes no sense to write down an equation involving $x+t$ because this equation would depend on our definition of "second" and "meter", and there is no reason why the laws of physics should depend on the second being defined to be 9,192,631,770 times the period of some radiation mode of a cesium atom. But, if we choose to work in natural units, then this is not an arbitrary choice because, as the name suggests, natural units are uniquely determined given fundamental constants of physics. In natural units, there is nothing wrong with writing an equation involving $x+t$, because we remember that we have made a special choice of units, and the equation will hold only in those units.
Of course, any equation that you can write in natural units can still be translated into arbitrary units. Take Einstein's famous mass-energy equivalence. In natural units ($c=1$) it states that $E = m$. Obviously, in arbitrary units, this is a bad equation because if $E$ is measured in Joules, and $m$ is measured in kg, then it would depend on the definitions of Joules and kg. But that's fine, because this equation only holds in natural units. Its translation into arbitrary units is $E = mc^2$, and the units now match up.
Can't add covariant 4-vectors and contravariant 4-vectors. Again, this is because in special relativity, in order to write down components of vectors we have to make an arbitrary choice of coordinate directions in space-time. Equations we write down in special relativity shouldn't depend on this choice, and this prevents us from adding covariant 4-vectors and contravariant 4-vectors because they transform differently when you change coordinate directions in space-time.
Can't add things in different vector spaces. This is just because, if $\mathbf{v}$ is in one vector space and $\mathbf{w}$ is in a totally different vector space, then you wrote an equation involving $\mathbf{v} + \mathbf{w}$ then it would depend on how you relate the bases between the two vector spaces, which -- given that they are totally different spaces -- there is no way to do non-arbitrarily.
One definition of work is "a change in energy." Any change in a physical quantity must have the same units as that quantity.
Different kinds of work are associated with different kinds of energy: conservative work is associated with potential energy, non-conservative work with mechanical energy, and total work with kinetic energy. In fact, that's one way to see the oft-quoted Law of Conservation of Energy:
$$
W_{total}=W_{non-conservative}+W_{conservative}\\
\Delta KE=\Delta E - \Delta PE \\
\therefore \Delta E=\Delta KE + \Delta PE
$$
So just like impulse (which is a change in momentum) has the same units as momentum, work has the same units as energy. Any change in a physical quantity must have the same units as that quantity. A change in velocity has units of velocity, etc.
A more difficult question might be why torque has the same units as energy. This is more subtle, but the key concept is this: units are not the only thing that determines a quantity's interpretation. Context matters too. Energy and torque may have the same units, but they are very different things and would never be confused for one another because they appear in very different contexts.
One cannot blindly look at the units of a quantity and know what is being discussed. A dimensionful quantity might be meaningless or meaningful depending on the context, and it's meaning can change with that context. Action times speed divided by length has the same units as energy but without any meaningful interpretation (as far as I'm aware).
Best Answer
Zasso pointed it already out:
Scaling up a ant to human size means volume (weight) increasing by length proportional $l^{3}$, but the force of muscles is determined by cross section (not muscle weight), so muscle force goes proportioal to $l^{2}$.
Smaller factors are likely:
i did some quick further search on "robot insects" on this interesting topic. This article is quite worth reading and relating biological to technological limits as well as current state of the art in nanobionics:
short example as Sonny asked for in comment:
ant with 10 mm length & 10 mg mass
$\Rightarrow$ lets scale up to human size (2m) $\Rightarrow$ means a factor of 200. So the mass scales with 200x200x200=8000000 (Volume $\propto$ $l^{3}$ ) $\Rightarrow$ human sized ant=80 kg. But muscle forces scales only by factor 200x200=40000. The small ant can carry 100x10mg of her own mass=1g, the human sized ant should be able to carry 1g x 40000=40 kg.
Conclusion: pretty comparable to a avg. 80 kg human man able to carry 40 kg!