How do I find the frequency of a particular bin in a cosine transform, and two other questions
DCTs (they exist under different flavors) map data along some cosine-related frequency axis, from $0$ to... the maximum frequency of your data, which we don't know, since we don't have its sampling frequency. So the realtive frequency index is implicit, between $0$ and $255/256$ in relative frequency. $2.669825 $, mostly probably, is the DC component, here the average of your signal, divided by $\sqrt{N}$, where $N$ is your number of samples ($256$). If I try to reconstruct your data with a type-II DCT, here is what I get:



The so-called "Frequency index" should be taken with caution, as it really depends on the DCT flavor, as commented be RBJ.

I'm not sure the signal is exactly yours, as I did a guess on the DCT type. To filter above $10$ Hz, you can see to zero the amplitude of all frequency indices above some ratio related to your sampling frequency. DCTs are discrete transforms, acting on signals with "unit" sampling period. So without data sampling information, you cannot get Hertz from the DCT spectrum alone, unless you know spectral characteristics on your signal that can be detected in the spectrum.

Negative values simply mean that the cosine contribution at that frequency is negated. 

But if you want to filter data, using a discrete Fourier transform is probably more appropriate.