It looks like you might not have been careful enough in defining your coordinate system. Are you defining the velocity positive downward or positive upward?
Also, in part (a), you didn't divide the 10 N by the mass, though you divided everything else by it. I'm guessing you were just sloppy with your parentheses in your second step. It should have been
dv/dt = (10+0.5v)/m. It looks like you fixed that problem in the next line.
You didn't separate out the variables correctly. The next step after the one I just mentioned should have been
dv = (20+v) dt.
I see that, again, in your next step you corrected the mistake. You really have to be a lot better with your parentheses! Don't write so that people can understand you. Write so that no one can misunderstand you!
Your solution, v = 40 e^t - 20 cannot possibly be correct. An exponentially increasing velocity? Surely common sense would argue that that is an impossibility when the only two known forces are opposing the direction of motion. So, I would go back to your original DE, and carefully check out the signs. Your general idea is correct, but I think you have a sign error, or maybe errors, in there somewhere. Do the same for the rest of the problem.