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Temptative answer. If we call f(t) the fraction of the distance covered at time t. Then f(0)=0 and f will be positive soon after take off. So initially f grows. I guess f will grow for ever. So either f will reach a limit somewhere and the rocket will hang at say 1/3 between earth and the galaxy. Or the rocket will smash its goal. which of the two will happen?

Second question. The speed due to expansion is proportional to the distance in between. This suggests that distance and speed increase exponential. So even an initial speed of c/2 doesn't guarantee the rocket in the first question to reach the galaxy because the galaxy might be running even faster before the rocket comes close. Is this reasoning correct?

Thanks a lot for your answers!