Intermediate macroeconomics: optimal bundle for quasilinear utility?
From the utility function it seems L is leisure.

a) I think you went about a) in a wrong way. For a, simply you want to work as little as possible but be able to pay 11 in taxes. So you will want to earn exactly 11USD and not more (earning more means working more, which you want to minimize). So at 0,50USD an hour you have to work 22 hours. Since h=24 the maximum leisure you can have is 24-22=2.

b) Where you are going wrong in your analytical approach for b) is not including the budget constraint in your problem (when taking derivatives), which you should as you explained for the graphic analysis. 

For consumption you have expenditures - income: p*c=wage*labor - T (technically an inequality, but in this case you can assume it holds with equality). I would suggest using a langrangian with this constraint. You also have the constraint that Leisure=24-labor. This you can plug in directly and is advisable so that your problem only has two unknowns (c and labor).
Solving this problem now should do the trick.