We define the Kolmogorov Complexity to be independent of any
particular programming language for bit string x as the length of the shortest
string <M,w> where TM M on input w halts with x on its tape
and <M,w> is some specified fixed encoding of the pair {M,w}.
Yes, generally $C(111...1) \leq C(1010...10)$. You can see why it couldn't be more noting that you can apropriatly change a program $M$ or a input $w$ for $10...10$ making substitutions of zeros for ones (intuition: repeat "01" k times-> repeat "11" k times). This don't change the size of neither $M$ nor $w$. And eventually it will be less, for in general programs for the first are simpler then programs for the second.