f(t) can have any units you like. It's just a time varying function.

F(0) is going to have the same units as F(s) for any s. F(0) is just the integral of f(t) over time between 0 and infinity.

This adds a factor of time to your units of f(t). So if f(t) is in units U, F(s) will be in units (U seconds)

If you don't like using F(0) like this you can note that the complex exponentials in the integral are unitless and obtain the same result.

I have to admit I never ran into a case where we had to be concerned about the units of the transform.