Why can't you add apples and oranges, but you can multiply and divide them?

Apples and oranges are actually a rather bad example. The reason why it doesn't make sense to add quantities with different dimensions, but it does make sense to multiply (or divide) them is scale invariance.

Let U be the unit of some quantity $u$, and $V$ be the unit of another quantity $v$. Now say we change the scale of U, i.e. we instead use a different unit U' such that $1U = 10U'$. For $V$ we do the same, only that there we choose $V'$ such that $1V = 5V'$. If we compute the sum $s$ of $u$ and $v$ in units U,V we get $$ s = u + v $$ If, instead, we compute the sum in units $U'$ and $V'$, however, we get $$ s' = 10\cdot u + 5\cdot v $$ Note that $s$ and $s'$ don't just differ by a factor, i.e. we can't convert $s$ from unit $U+V$ to $s'$ in unit $U'+V'$ without knowing the original values of $u$ and $v$.

Compare this to the situation of a product. If we compute the product $p$ of $u$ and $v$ in units $U$ and $V$, we get $$ p = u\cdot v $$ If, instead, we compute it in units $U'$ and $V'$, we get $$ p' = (10\cdot u) \cdot (5\cdot v ) = 50\cdot p \text{.} $$ So $p'$ is simply $p$, expressed in a different unit P', with $1P = 1UV = 50P' = 1U'V'$.

So why do you want scale invariance? We want that, because the scale of physical units is usually completely arbitrary. There's nothing fundamental about 1 meter, or 1 inch, or 1 Volt - we just picked some reference value. But since the reference value is arbitrary, the actual physics must not change if we replace it by a different one. Which it doesn't, so long as we only multiply and divide, but not add or subtract values with different units, as the example above shows.

And this is also why apples and oranges are a bad example. We don't expect scale invariance for these, because apples and oranges are discrete objects, so there's a canonical definition of what "1 apple" means. So adding apples and oranges makes perfect sense, and we may e.g. assign the result the unit fruits.

I suppose you can add apples and oranges. Just take the external direct sum of the fruit spaces. Moreover, you may recall Feynman's theory of everything: take the equations of physics and write them homogeneously $E_1=0, E_2=0, \dots$ then the theory of everything is simply: $$ 0=E_1=E_2= \cdots $$ you might note this is dimensionally inconsistent.

The addition question is related to the distributive property of numbers: $ax+bx=(a+b)x$

With: $$\begin{align}3\,\mathrm{oranges}+5\,\mathrm{oranges}&=(3+5)\,\mathrm{oranges}\\&=8\,\mathrm{oranges}\end{align}$$

And the lack of any such property for an expression like: $$\begin{align}3\,\mathrm{oranges}+5\,\mathrm{apples}\end{align}$$

Meanwhile with multiplication, the associative and commutative properties permit this: $$\begin{align}(3\,\mathrm{oranges})\cdot(5\,\mathrm{apples}) &=(\mathrm{oranges}\,3)\cdot(5\,\mathrm{apples})\\ &=\mathrm{oranges}\,(3\cdot(5\,\mathrm{apples}))\\ &=\mathrm{oranges}\,((3\cdot5)\,\mathrm{apples})\\ &=\mathrm{oranges}\,(15\,\mathrm{apples})\\ &=(\mathrm{oranges}\,15)\,\mathrm{apples}\\ &=(15\,\mathrm{oranges})\,\mathrm{apples}\\ &=15\,(\mathrm{oranges}\,\mathrm{apples})\\ &=15\,\mbox{orange-apples}\\ \end{align}$$

So from one perspective, the question is about which properties of arithmetic we use, and why isn't there a property like $ax+by=(a+b)(x+y)$. And there is no such property because it doesn't even hold with integers, with integer multiplication defined as repeated addition.

I think you should look at this symbolically. The units are just unevaluated "dummy" variables that stay symbolic (because they represent a physical quantity they can't be given a numerical value). This is nothing special, we also like to keep $i$ and $\pi$ unevaluated even when they have a bit more mathematical properties defined than our units do.

When you are computing with units, you are leaving them unevaluated - both in summation and multiplication! Think about it:

$$3\text{orange}\times 4\text{apple}=12(\text{orange}\times \text{apple})$$

The multiplicative group that units belong to may define an alias for this particular product, but that's just a substitution rule (like joule=newton*meter). So, in essence, you are not multiplying oranges and apples, you are leaving the product unevaluated. The same goes for summation: $$3\text{orange}+4\text{apple}=3\text{orange}+4\text{apple}$$ It's just that you can't simplify into a product of value*unit. Because we expect final expressions to be in the form value*unit, we say we can't do that, but above expression by itself is mathematically valid (although physically nonsensical, because there is no physical quantity with units 3orange+4apple, or, 0.75orange+apple if you want).

The same goes for evaluating pure mathematical functions on values with units! For instance, $\sin(40\text{apple})$ is perfectly valid expression, but it's irreducible. It has to be left in this form, because there is no numerical value that we can substitute into $\text{apple}$. However, this is somewhat alleviated with logarithms. It's common in physics to get intermediate expressions of the form $$\log V_1-\log V_2=\beta t$$ or something along these lines. $\log (\rm m^3)$ of course doesn't have a numerical value, it's just an irreducible symbolic entity. However, logarithms have a nice property of converting product into summation, so the problematic symbolic entity cancels out (producing $\log\frac{V_1}{V_2}$ which is a pure function evaluated on a pure number).

As soon as a unit can evaluate to numerical value, the "problem" disappears. For instance, the degree is simply ${}^\circ=\frac{\pi}{180}$ and percent is $\%=\frac{1}{100}$ and radians are just $\text{rad}=1$, so you can write $\sin(45^\circ+50\%+2\text{rad}+5)$ and there are no problems with summation of different units whatsoever.

To sum up: units are quantities that by definition don't need to evaluate to numerical values (they are "handles" that point to the physical world). We treat the unevaluated product of value and unit as valid but not sums of mismatched units simply because the first can be reinterpreted back into the physical world, while the second usually have no reasonable meaning. Mathematically, there's no difference.