What are some functions that satisfy:
1) $x(t) \to y(t)$ and then $ax(t)\to ay(t)$
but not this property:
2) If $x_1(t) \to y_1(t)$ and $x_2(t) \to y_2(t)$ then $x_1(t) + x_2(t) \to y_1(t) + y_2(t)$
I cannot think of any. I considered $y = x^2$, but that doesn't satisfy the first property. Nothing else seems possible. Also, this question came from my professor, but Wikipedia says that homogeneity is defined to be: If $x(t) \to y(t)$ then $ax(t) \to a^ky(t)$ which would make more sense to me.
Here is an example function from $\mathbb R^2$ to $\mathbb R$ that satisfies homogeneity,
$$f(a\cdot x)=a\cdot f(x)$$
for all $a\in\mathbb R$, $x\in\mathbb R^2$, but does not satisfy additivity,
$$f(x+y) = f(x)+f(y)$$
for all $x,y\in\mathbb R^2$.
Define
$$f((r,s)) = \min(r, 2s)$$
Then it is easy to see that this function satisfies homogeneity, but letting $x=(1,0)$, $y=(1,1)$ we see that
$$f(x)=\min(1,2\cdot 0)=\min(1,0)=0$$ $$f(y)=\min(1,2\cdot 1)=\min(1,2)=1$$ but $$f(x+y)=f((2,1))=\min(2,2\cdot 1)=\min(2,2)=2$$
So here, $f(x+y)\ne f(x)+f(y)$.
Note that the basis for this example is in a section of the very Wikipedia article that you reference in your question.
If the $a$ range over the rationals, I can think of a similar example of a real-valued function of real numbers, but it is more complicated to explain.