Use of approximation sign

You can chain any binary relations. For example, $$ a\leq b=c<d\geq e\approx f $$ means that $a\leq b$, $b=c$, $c<d$, $d\geq e$, and $e\approx f$. Each binary relation only describes the relation between the two numbers (or what have you) around it. It does not mean that $a\leq f$ or $a\approx f$ or anything like that.

From the above chain you can conclude, for example, that $a<d$, but you can't compare $c$ and $f$ without more information. What you can conclude about $a$ and $f$ depends on all the symbols in between.

For another example, $$ g=h\approx i=j=k\approx l $$ does indeed imply that $g\approx l$.

I would argue that two approximations is still usually fine. The relation "$\approx$" is not rigorously defined, and both the question and this answer are outside the realm of fully rigorous mathematics. It is defined heuristically (we kind of know what it means), and this heuristic meaning has properties. One of the problematic properties is transitivity. If stretched too far, it becomes ridiculous: $1.00\approx1.01\approx1.02\approx\cdots\approx1.99\approx2.00$ and so $1\approx2$, and by induction $n\approx m$ for any two integers $n,m$. However, I don't see an issue in practical calculations as in the question, and I would confidently conclude $g\approx l$ in my second example. The approximation gets potentially worse when repeated, but just a couple of steps is no big issue in this context.

It is important to tell your students explicitly and repeatedly that $$ 129.87\cdot 6\approx 130 \cdot 6 = 780 $$ means two statements in one: $129.87\cdot 6\approx 130 \cdot 6$ and $130 \cdot 6 = 780$. My experience suggests that not all students will understand this unless (even if?) told. It follows from these two statements that $129.87\cdot 6\approx780$, but this is not stated directly.

I would suggest your first option. But the second option isn't really wrong either; both $130 \cdot 6 = 780$ and $130 \cdot 6 \approx 780$ are true. But it can be confusing to use "$\approx$" when the two things are actually equal.

I would personally say that $130\cdot 6$ is actually equal to $780$, so an equals sign is appropriate. I don't have any concrete evidence to back up whether this is conventional, however. Probably because $\approx$ isn't the most common of relational symbols in mathematical texts.

On the other hand, when using $\leq$ and $\geq$, this type of notation is indeed standard. A line like $$ \text{expression }1 \leq \text{expression }2 = \text{expression }3 \leq \text{expression }4 $$is not uncommon at all, although there are definitely authors who would use only $\leq$ in the above line.

It is difficult to convey two meanings using only one symbol. I would write $$129,87\cdot 6\approx 130 \cdot 6 \mbox{ and } 130 \cdot 6 = 780$$ so $$129,87\cdot 6\approx 780$$

The "and" is a pointer that (a) first we approximate and (b) then we compute exactly. The final result is an approximation only because of (a).

Related to this, it is useful to tell students that $=$ is transitive, while $\approx$ is not. You may use simple examples (search under "sorites paradox") to nail the point.