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 questions refer to properties of explicit, implicit (+ Newton to solve the implicit nonlinear equations), or both ('IMEX', see e.g. here) time-stepping schemes. And my answers base on results from the numerical analysis single-step methods.
- If you use convergent schemes, in terms of the convergence order, there is no general advantage of using implicit schemes (see. 2.). However, for stiff systems, e.g. your system containing a Laplacian, there are implicit schemes that are stable without time-step restrictions. Nevertheless, in theory, for the explicit scheme, you get better results with smaller time-steps, as long as your equation itself is stable (e.g., referring to Picard-Lindelof Theorem, if $F_h$ is Lipshitz in the second argument) and your time-step is not too small.
- 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.) If you make the Newton error sufficiently small, you can still improve accuracy by decreasing the time-step or by using time-stepping schemes of higher order.
- The constant $C$ in the error estimate for the global error grows exponentially with the length of the time-interval. See, e.g., here 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.