Positive integers $k = p_{1}^{r_{1}} \cdots p_{n}^{r_{n}} > 1$ satisfying $\sum_{i = 1}^{n} p_{i}^{-r_{i}} < 1$

Say $k$ has the prime factorization $p_1^{r_1} p_2^{r_2} \cdots p_n^{r_n}$. Then let $H(k)$ be the "hyperbolicity" of this integer, $H(k) = \sum_{i=1}^n p_i^{-r_i}$. Hyperbolic integers are those integers $k$ that have $H(k) < 1$.

Now, $H(k)$ for some "random" integer $k$ looks like a sum of independent random variables $X_p$, one for each prime $p$, where $P(X_p = p^{-k})$ is the probability that a random integer is divisible by $p^k$ but not $p^{k+1}$. Thus $P(X_p = p^{-k}) = (p-1)/p^{k+1}$.

In particular $$ E(X_p) = \sum_{k \ge 1} p^{-k} P(X_p = p^{-k}) = \sum_{k \ge 1} {p-1 \over p^{2k+1}} = {1 \over p^2 + p} $$

Things get a bit tricky in finding the distribution of $\sum_p X_p$; in particular a central limit theorem won't work, because the contribution of each of the small primes is too large. But there is some limiting distribution, and I suspect it at least bears some resemblance to the limiting distribution of the actual hyperbolicities of integers.

Incidentally, my tests seem to point to a natural density of hyperbolic integers somewhere in the neighborhood of 99 percent. I'd conjecture that integers with hyperbolicity less than $k$ have a natural density. There are similar results on the distribution of $\sigma(n)/n$ where $\sigma(n)$ is the sum of divisors of $n$; see for example Marc Deleglise, Bounds for the density of abundant integers.

[[EDIT: Let $ k = p_1^{r_1} \cdots p_n^{r_n} $ as in your post. Then define the hyperbolicity $ H(k) = \sum_{i=1}^n p_i^{-r_i} $.]]

We can refute the idea in your second question in a much more general manner than you might expect. Note that since $ \zeta_P(1) $ diverges, numbers can have arbitrarily large hyperbolicity; similarly, since $ H(2^n) = 2^{-n} $, it follows that numbers can have arbitrarily small hyperbolicity. Define $ d(x) $ to be the asymptotic density of natural numbers satisfying $ H(n) > x $. We will demonstrate the strict inequality $ d(x) > 0 $ holds for all $ x > 0 $, and even deduce a satisfying lower bound for sufficiently, erm... not-small $x $.

Fix $ x \in (0,\infty) $. Let $ k $ be the smallest natural number such that $ 1/2 + 1/3 + 1/5 + \dots + 1/p_k > x $. Then for $ n = 2(3)(5)\cdots (p_k) $, $ H(n) > x $. Note that hyperbolicity, taken as an arithmetic function, is multiplicative, though not completely. In fact, by categorizing which prime factors belong where and when they're duplicated, one can see the following formula:

$$ H(ab) = H(a) + H(b) - H(\gcd(a,b)^2) + H(\gcd(a,b)) $$

This isn't necessary for our argument, I'm just putting that out there for posterity.

At any rate, we may conclude that for any number $ m $ which is coprime with $ n $, we can obtain another sufficently hyperbolic number via multiplication: $ H(nm) > H(n) > x $. Note that the density of numbers coprime with $ n $ can be expressed as

$$ y = (1-1/2)(1-1/3)(1-1/5)\cdots (1-1/p_k). $$

This formula echoes the Sieve of Eratosthenes: it follows from the fact that the probability a number is divisible by a prime is independent of the probability it is divisible by a different prime.

Finally, if you take the set of all numbers $ m $ coprime with $n $, and multiply them all by $ n $, they'll become $ n $ times more sparse, hence the density of all numbers of the form $ n m $ is $ y / n $. This proves $ d(x) \ge y/n $, which is a pretty good lower bound so long as $ x $ isn't too small (this is my subjective impression). For $ x = 1 $, as in your original question, we have the close lower bound $ (1/30)(1/2)(2/3)(4/5) $ or about 0.89%.

So far I haven't been able to determine whether or not there exists any $ \epsilon > 0 $ such that $ d(\epsilon) = 1 $. If there is one, then there would exist a maximum $ u $ such that $ d(\epsilon) = 1 $ for any $ \epsilon \in (0,u) $, which would make an interesting new constant. Note that $ d(x)$ is decreasing, though I'm not sure if it's continuous. What would really be interesting is if $ d(x) $ were differentiable on some interval.