Put
$$
B= C \frac{\log (10\sigma(n)/n)}{\log \log (10 \sigma(n)/n)}
$$
for a suitably large positive constant $C$. Then I claim that the desired inequality holds with
$$
D = \frac{n}{(\log n)^B},
$$
for all large $n$. Since $\sigma(n)/n \ll \log \log n$, it follows that one may always take
$$
D = n \exp\Big( -C \frac{\log_2 n \log_3 n}{\log_4 n}\Big), \tag{1}
$$
with $\log_j$ denoting the $j$-th iterated logarithm, and in fact this is in general the best possible. If $\sigma(n)/n$ is bounded by $1/\epsilon$ (as in Question 2) obviously one can obtain a stronger result, as $B =C \log(1/\epsilon)/\log \log (1/\epsilon)$ is now permissible.

Now for the proof. Put
$$
A = \frac 12 \log \log (10\sigma(n)/n), \text{ and } \alpha= \frac{A}{\log \log n}.
$$
Note that (writing $d=n/k$)
$$
\sum_{\substack{ d|n \\ d\le D}} d =n \sum_{\substack{k|n \\ k > (\log n)^B}} \frac{1}{k} \le n (\log n)^{-B\alpha} \sum_{k|n} \frac{1}{k^{1-\alpha}} = ne^{-AB} \sum_{k|n} \frac{1}{k^{1-\alpha}}. \tag{2}
$$
For large $n$ we have
$$
\sum_{k|n} \frac{1}{k^{1-\alpha}} \ll \prod_{p|n} \Big(1 + \frac{1}{p^{1-\alpha}}\Big) \ll \exp\Big( \sum_{p|n} \frac{1}{p^{1-\alpha}}\Big).
$$
To estimate this, divide the primes $p|n$ into three ranges: $p\le (\log n)^{1/A}$, $(\log n)^{1/A} \le p \le (\log n)$ and $p> \log n$. The contribution of the first range is
$$
\sum_{\substack{ p|n \\ p\le (\log n)^{1/A}} } \frac{e}{p} \ll \log (\sigma(n)/n).
$$
The second range gives
$$
\le \sum_{(\log n)^{1/A} \le p\le (\log n)} \frac{1}{p^{1-\alpha}} \le e^A \sum_{(\log n)^{1/A} \le p\le (\log n)} \frac 1p \ll e^A \log A.
$$
The final range gives (since the number of distinct primes dividing $n$ is $\ll \log n/\log \log n$)
$$
\ll \sum_{\substack{p|n \\ p>\log n}} \frac{1}{p^{1-\alpha}} \le \frac{1}{(\log n)^{1-\alpha}} \sum_{p|n} 1 \ll \frac{e^A}{\log \log n}.
$$
Combining all these bounds, and using it in (2) we find
$$
\sum_{\substack{d|n \\ d\le D}} d \ll n e^{-AB} \exp\Big( O(\log (10\sigma(n)/n))\Big) \le n,
$$
upon taking $C$ suitably large. This completes the proof that the claimed choice for $D$ works.

Let me now quickly say why the general form (1) is optimal. Take $n$ to be the lcm of all the integers up to some point (which is roughly $\log n$ by the prime number theorem). Then, roughly speaking,
$$
\sum_{\substack{k |n \\ k >(\log n)^u}} \frac{1}{k} \gg \sum_{\substack{ (\log n)^{u+1} \ge k \ge (\log n)^u \\ p|k \implies p\le (\log n)}} \frac 1k \gg \rho(u+1) \log \log n,
$$
where $\rho$ denotes the Dickman function. That is, the divisors of $n$ basically correspond to $\log n$ smooth numbers, and I have invoked the asymptotic for smooth numbers here. Since $\rho(u)$ behaves like $u^{-u}$ we see that $u$ has to be about as large as $\log_3 n/\log_4 n$ in order for $\rho(u+1) \log \log n$ to become negligible. This shows that the range in (1) cannot be improved (apart from the constant $C$).