Log log regression with fixed effects and cross elasticity of demand
Note that you have an identification problem here because price is an endogenous variable. A way to deal with this problem would be to find an instrument (e.g. prices  of the same good in other markets as in Hausman (1996)).
Regarding your questions.

Yes (although fixed effects are probably not the right wording here).
Yes.
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.
I would think about the status of these types. Is the same product with different attributes? Are these types substitutes? Is it a kind of aggregation? Regarding interpretation: you interpret one category with respect to another (or to a benchmark category in case of more than one category).