Find highest frequency of a very manky signal
I think most of your questions can be solved by subtracting then mean from the signal. Namely the all sinusoidal waves, with a nonzero frequency, have a mean of zero. So when the mean of a signal is not close to zero, then it will show up at 0 Hz (you can look at this as $\cos(0\,t)=1$).

After a closer look and some testing by myself it does seem that only removing the mean would not fix your problem. Your signal seems to be contaminated by white noise, which has been integrated probably twice (so low frequencies are much more present than high frequencies). You could try to subtract a higher order polynomial fit, so linear or even quadratic. These fits can easily be obtained using some linear algebra, for example for a quadratic fit:

X = [ones(N, 1), t', t'.^2];
b = (X' * X) \ X' * y';
z = y - (X * b)';


Or you could also look at the biggest peaks above $f^{-2}$, with $f$ the frequency vector. However if you have a signal with normal white noise you should probably just stick with removing the mean, since my proposed methods try to make use of the fact that the present noise is white noise which has been integrated twice.