I assume, that you have conducted a space discretization, so that you are about solving the (vector-valued) ODE
$$
   \dot u_h(t) = F_h(t,u_h(t)), \text{ on [0,T] }, u_h(0) = \alpha.
$$
Then your question refer to properties of *explicit*, *implicit* (+ Newton to solve the implicit nonlinear equations), or *both* ('IMEX', see e.g. [here][1]) time-stepping schemes. And my answers can base on results from basic numerical analysis.

 1. If you use convergent schemes, then you get better results with smaller time-steps, as long as your equation itself is stable (e.g. if F_h is Lipshitz) and your time-step is not too small.
 2. You can find examples, where explicit schemes perform better. (Theoretically, you can reverse the time in your example, start from the terminal value, and find implicit and explicit interchanged.) You can improve accuracy by decreasing the time-step or by using time-stepping schemes of higher order.
 3. The constant $C$ in the error estimate for the global error grows exponentially with the length of the time-interval. See, e.g., [here][2] for the explicit Euler scheme. This is true for every single-step method. As the estimate is of type $err \approx C \tau^p$, $p>0$, a smaller time-step $\tau$ only postpones this effect. 


  [1]: http://eprints.maths.ox.ac.uk/1194/1/NA-03-14.pdf
  [2]: http://en.wikipedia.org/wiki/Euler_method#Global_truncation_error