Log log regression with fixed effects and cross elasticity of demand

I have a time series of units sold, and price. I'd like to calculate elasticity of demand wrt to price and a few other variables, some of them are fixed effects.

Qp = b0 + b1 * log(Price) + b2 * Location + b3 * log(Income) + b4 * log(HouseHolds)


I am using log-log regression to calculate the elasticities, and the independent variable of interest is Price. Qp is # of units sold.

My questions are

1. Can I use Median Income, and Households are fixed effects for the time period in log-log regression model.

2. Interpretation of Log - Log regression. 1% change in price changes the units sold by b1%(coefficient of price), Ceteris paribus, all else being equal.

3. Cross Elasticity. What I add a term for # of available substitutable units? How would I interpret the term?

Qp = b0 + b1 * Price + b2 * Location + b3 * Income + b4 * HouseHolds + b5 * substitutable units

4. I also have additional categorical variable, i.e type of unit sold. type 1 and type 2. Does it make sense to include that variable in the equation or fit a regression separately for each type? How would you interpret different coefficients of categorical variable?

5. I'd like to find the point where increasing the Price negatively affects units sold.

3. If you want to capture a cross elasticity effect you need to introduce the own price as well as price of substitutes into the equation. You may face a dimensionality problem due to the large number of parameters to be estimated. The effect substitutable units would be interpreted as the effect of the availability of substitutes on demand.