Checking equality of integers: O(1) in C but O(log n) in Python 3?
Integers are just binary strings and, to determine equality, both languages will compare the strings bit-by-bit.

Not quite. C ints are machine-word-sized and compared with a single machine instruction; Python ints are represented in base $2^{30}$ (see e.g. https://rushter.com/blog/python-integer-implementation/) and compared digit-by-digit in that base. So the relevant base of the logarithm is $2^{30}$.
If at least one of numbers can be bounded by $2^{30d}$ for any fixed $d$, the comparison is $O(1)$ (because the number of digits is compared first), and if they can't, other operations are likely of much more concern than equality comparison. So in practice I'd say it's very unlikely to matter and if it does you'll know (and would be using not ints but something like the GNU Multiple Precision Arithmetic Library in C as well).