This answer adresses the first of the two questions (and was originally written, except for minor changes, for a slightly vaguer version of the question; thus the material that might not seem completely fitting now, except that it could be helpful for the second question, so I leave it).

It is a result of Lindström (On the Binary Digits of a Power, Journal of Number Theory, 1997) that, using his notation and denoting by $B(\cdot)$ the numer of 1s in the binary expansion in other words the sum of digits,
$$
\limsup_{m\to \infty} \frac{B(m^h)}{\log_2 m}= h
$$

which means precisely that for fixed $c$ the supremum is indeed $1$; indeed it works for any exponent not just powers of $2$.

I have not studied Lindström's proof in detail, but it seems to be explicit so that one could derive information for the $x_0$; and for squares the paper by Dromota and Rivat contains another explicit construction that could also be used.

Even more generally, the same is true for every polynomial of degree $h$ (with integer coeficients and positive leading coefficient).

Yet, it is true that such numbers, powers with many $1$, are in a certain sense rare.

There are various further (in part recent) results around this question. For example:
$$
\frac{1}{N} \# \left\{ n \lt N \colon B(n^2) \le \log_2 N + y \sqrt{\frac{\log_2 N}{2}} \right\} = \Phi(y)+o(1)
$$
where $\Phi$ denotes the normal distribution function. In other words, this $B(n^2)$ behaves like the sum of $2 \log_2 N$ independent random variables $0$ and $1$ with equal probability.
This result is a special case of a result of Bassily and Kátai (Distribution of the values of q-additive functions on polynomial sequences, Acta Math. Hung. 1995)

For other results related to this, and the above mentioned information in more detail and nice constructions related to the above phonomenon, see for example Drmota and Rivat "The sum of digits function of squares" (Journal LMS, 2005) ; for similar investigations for arbitrary polynomials and q-ary digits see this more recent paper by the same authors and Mauduit

Or, for investiagtions of the ratio of $B(m^h)/B(m)$ see this recent paper by Hare, Laishram, Stoll "Stolarsky's conjecture and the sum of digits of polynomial values"

(Added: I do not know if anything on the second question is known or could be derived from known things; the distribution result seems to support what Aaron Meyerowitz says, it could be worth looking into the results on the ratio I mentioned.)