How do we assign length to dimensions in string theory?

Let me change the order of your questions a bit:

About considering them as tiny, couldn't they be just as infinite as the other three but because everything currently known to us is at most 3D we have no idea of what could be going on in other spatial dimensions?

The existence of extra dimensions affects the physics that we see in our 3+1 dimensional world, and we have no experimental evidence for such infinite dimensions. However, if the extra dimensions are tiny enough we will not be able to detect them with current experiments.

I should note that there are theories of large extra dimensions, where the Standard Model is located on and restricted to a 1+3 dimensional subspace (a brane more precisely), but even then gravity would still propagate in these large extra dimensions and alter the physics we observe.

So how do we assign length to those "extra" dimensions?

Since large extra dimensions are not observed, one compactifies the extra dimensions. As the simplest example, a single extra dimension could be curled up as a circle, which has a characteristic length given by the radius $R$. In most scenarios this length would be of the order of the Planck length. In general to obtain models closer to that of the Standard model, the extra dimensions are compactified on more complicated geometries (Calabi-Yau manifolds).

To the extent of our knowledge, could the universe contain more dimensions that have very little to do with space or time? If yes, do we still have to consider those extra dimensions as spatial?

I don't think this makes much sense. You want to add dimensions, which are then NOT dimensions? So in the end you are asking for the consequence of something you haven't defined. To add to this, the dimensions of string theory are not just something you add by hand as you like. They are required for consistency of the theory.