Is the property "being a derivative" preserved under multiplication and composition? Since differentiation is linear, we therefore have that if $f, g: I\to \mathbb{R}$ is a derivative (where $I\subset \mathbb{R}$ is an interval), then so does their linear combination. What if we consider their multiplication and composition?
Due to the forms of the product rule of differentiation of product function and chain rule of differentiation of composition, I highly doubt their product or composition necessarily is still a derivative, but I cannot construct counterexamples.
 A: Let $X$ be the set of functions from $\mathbb R$ to $\mathbb R$ that have antiderivatives on all of $\mathbb R.$
Claim: There exists $f\in X$ such that $f^n\notin X$ for $n=2,3,\cdots.$ (Here $f^n$ is the $n$th power of $f,$ not the $n$th iterate.) This addresses both the product question and the composition question. If the claim is true, then $f\in X$ but the product $f\cdot f \notin X.$ Also, with $g_n(x) = x^n, n = 2,3, \dots,$ we see $g_n\circ f\notin X$ even though the $g_n$'s are very nice and $f\in X.$
The following lemma will be helpful:
Lemma: Suppose $f:\mathbb R\to \mathbb R,$ with $f=0$ on $(-\infty,0]$ and $f$ continuous on $(0,\infty).$ Then $f\in X$ iff $\int_0^x f$ converges for all $x>0$ and
$$\lim_{x\to 0^+}\frac{1}{x} \int_0^x f = 0.$$
Here the convergence of $\int_0^x f$ is in the improper integral sense, meaning $\int_0^x f = \lim_{a\to 0^+}\int_a^xf$ assuming the limit is finite.
Proof: $\implies:$ Let $f\in X,$ with $F'=f$ on $\mathbb R.$ Let $x>0.$ For any $a\in (0,x),$ $F\in C^1[a,x].$ Thus
$$\tag 1 F(x)-F(a) = \int_a^x f.$$
As $a\to 0^+,$ the left side of $(1)$ $\to F(x) - F(0)$ by the continuity of $F.$ It follows that the right side has a finite limit, hence $\int_0^x f$ converges. Note that we have shown
$$F(x)-F(0) = \int_0^x f.$$
Now $F'(0) = f(0)= 0.$ It follows that
$$0 = \lim_{x\to 0+}\frac{F(x)-F(0)}{x-0}= \lim_{x\to 0+} \frac{\int_0^x f}{x}$$
as desired.
$\impliedby:$ Simply define $F(x) = \int_0^x f.$ By the continuity of $f$ on $\mathbb R\setminus \{0\},$ we have $F'(x) = f(x)$ for $x\ne 0.$ At $0,$ we easily see the derivative from the left of $F$ at $0$ is $0.$ From the right we have
$$\lim_{x\to 0+}\frac{F(x)-F(0)}{x-0}= \lim_{x\to 0+} \frac{\int_0^x f}{x} = 0.$$
This shows $f\in X$ and the proof of the lemma is complete.
Proof of the claim: We build a slightly crazy $f\in X.$ For $n=2,3,\dots$ let $a_n = 1/n, b_n = 1/n + 1/2^n.$  Then the intervals $[a_n,b_n]$ are pairwise disjoint.
On each $[a_n,b_n]$ let $f$ be an isoceles triangular spike from the end points of height $2^n/n^3.$ Define $f =0$ everywhere else. Then $f$ satisfies the hypotheses of the lemma. Now
$$\int_0^1 f = \sum_{n=2}^{\infty} \frac{1}{2}\cdot \frac{1}{2^{n+1}}\cdot \frac{2^n}{n^3} = \sum_{n=2}^{\infty} \frac{1}{4n^3} <\infty.$$
To show $(1/x)\int_0^x f \to 0,$ suppose $x\in [a_n,a_{n-1}].$ Then
$$\tag 2 \frac{1}{x}\int_0^x f \le \frac{1}{a_n}\int_0^{a_{n-1}}f = n \sum_{k=n}^{\infty}\frac{1}{4n^3}.$$
The last sum is $O(1/n^2).$ So $(2)$ is no more than $nO(1/n^2) = O(1/n) \to 0.$ Thus the left side of $(2)\to 0$ as $x\to 0^+.$ The lemma now shows $f\in X.$
The easy part of the claim is showing higher powers of $f$ are not in $X.$ By the lemma, all we need to show is that $\int_0^1 f^2 = \infty.$ It will then follow that $\int_0^1 f^n = \infty, n=3,4,\dots $
Here's what I get (but the reader should check):
$$\int_{a_n}^{b_n} f^2 = \frac{2^n}{3n^6} .$$
Thus $\int_0^1 f^2 =\infty,$ by a mile. The proof of the claim is complete.
A: Let me address just one of your problems.

Problem. Suppose that $f$ and $g$ are both derivatives.   Under 
   what conditions  can we assert that  the product $fg$ is
  also a derivative?

The short answer is that this is not true in general.  In fact even if we assume that $f$ is continuous and $g$ is a derivative the product need not
be a derivative.  However if we strengthen that to assuming that $f$ is not merely continuous but also of bounded variation, then indeed the product with any derivative would be a derivative.
This is an interesting problem and leads to interesting ideas.
For references to the statements here and an in depth look at the problem here are some references:

Bruckner, A. M.; Mařík, J.; Weil, C. E. Some aspects of products of
   derivatives. Amer. Math. Monthly 99 (1992), no. 2, 134–145.
Fleissner, Richard J. Distant bounded variation and products of
    derivatives . Fund. Math. 94 (1977), no. 1, 1–11.
Fleissner, Richard J. On the product of derivatives. Fund. Math. 88
  (1975), no. 2, 173–178.
Fleissner, Richard J. Multiplication and the fundamental theorem of
  calculus—a survey. Real Anal. Exchange 2 (1976/77), no. 1, 7–34.
Foran, James 
  On the product of derivatives, Fund. Math. 80 (1973), no.
  3, 293–294.

I will edit in some links when I find them.  Foran and Fleissner were close childhood friends who ended up pursuing their PhD at the same time in Milwaukee.  Fleissner died in an automobile accident in 1983.
NOTE ADDED.  Elementary students are not going to want to pursue this topic to quite this depth.  But here is an exercise aimed at this level that they might find entertaining.

Exercise.  Consider the function $$f(x)=\begin{cases} \cos \frac1x, & x\not=0 \\ 0 &x=0 \end{cases} $$  Show that the function $f$ is a
  derivative but that its square $f^2$ is not.

A: From the product rule of differentiation or from the rule of partial integration, one obtains the following less known product rules:
\begin{equation}\int f(x)g(x)dx=f(x)G(x)-\int f'(x)G(x)dx\hspace{1.5cm}[Borman], [Goldstein], [Kowalk]\end{equation}
\begin{equation}\int f(x)g(x)dx=F(x)g(x)-\int F(x)g'(x)dx\hspace{3.9cm}[Mercer], [Reinsch]\end{equation}
$F(z)=\int f(z)dz$; $G(z)=\int g(z)dz$; $\gamma$: compositional inverse of the function $g$
From the chain rule of differentiation, the inverse rule of differentiation and the rule of partial integration, one obtains the following less known chain rules:
\begin{equation}\int f(g(x))dx=\int f(t)\gamma'(t)dt;\ t=g(x)\end{equation}
\begin{equation}\int f(g(x))dx=xf(g(x))-\int f'(t)\gamma(t)dt;\ t=g(x)\end{equation}
\begin{equation}\int f(g(x))dx=\left(\frac{d}{dx}F(g(x))\right)\int\frac{1}{g'(x)}dx-\int \left(\frac{d^{2}}{dx^{2}}F(g(x))\right)\int\frac{1}{g'(x)}dx\ dx\end{equation}
\begin{equation}\int f(g(x))dx=\frac{F(g(x))}{g'(x)}+\int F(g(x))\frac{g''(x)}{g'(x)^{2}}dx\end{equation}
This formulas show that integration rules for products and compositions of functions are possible only for some special kinds of component functions $f$ and $g$.
