Q1. Is there a sense in which the boundary operator $\partial$ is analogous to a derivative?
It may be worth noting that de Rham in (1936) and especially (1938, pp. 317–323) already spelled out most of the other replies’ analogies. Indeed the latter answers the title question with (my bold):
We already observed (...) an analogy between the boundary operation and differentiation: Both are linear, and performed twice, both give zero:
$$
\begin{align}
f(c_1\pm c_2)&=fc_1\pm fc_2,& f(fc)&=0,\tag1\\
d(\omega_1\pm\omega_2)&=d\omega_1\pm d\omega_2,& d(d\omega)&=0.\tag2
\end{align}
$$
This analogy actually goes much further (...). The differential of a product is given by the rule
$$
d(\omega_1\omega_2)= d\omega_1\omega_2+(-1)^p\omega_1d\omega_2,
%\quad
%\style{font-family:sans-serif}{\text{where }} p=
%\style{font-family:sans-serif}{\text{degree of }}\omega_1.
%\style{font-family:sans-serif}{\text{ ...}}
\tag3
$$
where $p=$ degree of $\omega_1$. (...) Our analogy’s starting point is the fact that chains, like forms, make an alternating ring (...) and to the differentiation rule for products corresponds the formula which gives the boundary of an intersection:
$$
f(c_2c_1)=c_2fc_1+(-1)^{n-p}fc_2\cdot c_1.
\tag4
$$
The formula is an exact match if we let the intersection $c_2\cdot c_1$ of two chains of dimensions $p_2$ and $p_1$ correspond to the product $\omega_1\omega_2$ of two forms of degrees $n-p_1$ and $n-p_2$. I followed the usual definitions, which is why one must reverse the order of factors. (...) To the index [sum of coefficients] of a $0$-chain there corresponds the number $I(\omega)=\smash{\int_V\omega}$ attached to any $n$-form, and one can show that all matching properties hold.
But I would now like to show that, much more than a mere formal duality, there is a deeper identity between $p$-chains and $(n-p)$-forms; or in other words, that on an $n$-dimensional manifold, a $p$-chain and an $(n-p)$-form are two instances of one and the same concept. This idea (...) is suggested by physical considerations, actually cases where physical quantities occur which can be represented, sometimes by a chain, sometimes by a form.
As @BS. said, de Rham goes on to unify both in the idea of a current $(c,\omega)$ (pre-distribution theory). Also, the formulas of @DenisSerre and @VaughnClimenhaga are ungraded versions of (4) and
$$
f(c_1\times c_2)=c_1\times fc_2+(-1)^{\dim c_1}fc_1\times c_2.
\tag5
$$
Both (4) and (5) are attributed to Lefschetz (1925, §1; 1926, 16.1, 49.2; cf. 1942, IV.2.3, IV.5.6, V.8.4) by e.g. van der Waerden (1929, p. 343; 1930, p. 129), de Rham (1932), Steenrod (1957, pp. 29, 40). But of these authors, only Steenrod uses $\partial$ and not some variant of $B$, $F$, or $R$. So...
Q2. What is the historical origin for the use of the symbol $\partial$ in topology?
Michèle Audin writes in Henri Cartan & André Weil: du vingtième siècle et de la topologie (2012, p. 46):
$d$ is for differential, no doubt, but what is $\partial$ for? Neither boundary nor frontier, in neither French nor German (nor even English), so? Well, Rand, in German, written in Gothic and abbreviated $\mathfrak{Rd}$, then $\mathfrak d$ (two letters, that was too much), then $\partial$ !
She doesn’t quote sources, but indeed we find in §16 of Seifert & Threlfall’s Lehrbuch der Topologie (1934), the following:
der Rand von $E^k$ [ist] hiernach die $(k-1)$-Kette
$$
\mathscr R\partial E^k = \varepsilon\sum_{i=0}^k(-1)^i(P_0P_1\cdots P_{i-1}P_{i+1}\cdots P_k)\tag6
$$
(...) Der Rand wird mit dem Symbole $\mathscr R\partial$ bezeichnet.
They still use the same notation in Variationsrechnung im Grossen (1938, p. 17 sq). The earliest I find using $\partial$ alone in combinatorial topology are Eilenberg (1938, p. 180) and Whitney (1937, pp. 38, 40, 46, 51). In point-set topology — your initial concern — it may also be Whitney: (1944, pp. 221, 247).
Question: has anyone anywhere seen the Gothic (Fraktur) spellings Audin alludes to?
