Part 1:

a. what will be the percentage change in quantity demanded if prices are increased by 50%:

Price Elasticity of Demand = % change in quantity / % change in price

-0.75 = % change in quantity / 50%
% Change in quantity = ( -0.75 * 50% ) = -37.5%

b. Will the total revenue of the firm increase, decrease, or remain the same when prices are increased by 50%:

There will be a decrease in quantity demanded by 37.5%.

Part 2: If there is a 20% across the board increase in income taxes, lowering the income of all consumers:

a. Will the sales of widgets increase, decrease, or remain the same?

Income elasticity of demand = % change in quantity demanded / % change in income.
As it is mentioned that widget is considered a luxury good so there will be a negative shift.

Price elasticity of demand = % change in quantity / 20%
2 = % Change in quantity 

% Change in quantity = 40%

b. By what percentage amount will the quantity sold change, if at all?

20% rise in income tax will reduce 40% demand.

Part 3: If gadget prices rise by 15%:

a. In what direction (increase, decrease, or no change) will widget sales change.

 Cross price elasticity of demand = % Change in Quantity demanded of A / % Change in Price of B

4 = % Change in quantity demanded of widgets / % Change in price of gadgets

4 = % change in quantity demanded of widgets / 15%
% Change in quantity demanded of widgets = 4 * 15
% Change in quantity demanded of widgets = 60%

b. By what percentage amount will widget sales change, if at all?

There will be an increase in demand for widgets by 60%