Multi-level models: xtmixed vs. xtreg, re

#1
Hi all,

I am dealing with a panel and multi-level data. Specifically, I have data (prices, etc...) related to numerous items (each identified by a unique id) within a product category in a given period. The measurements for the same item are repeated several times in this period: that is, for each item I have 20 observations each corresponding to a specific day. The items are selected from three different stores. Suppose initially that items are unique to a store, i.e. the same item is not in two or three stores, but only in one store.
My objective in this study is to study how some independent variables (e.g. brand, producer reputation, quality, etc...) affect the price of such product category. Moreover, I am very interest in the effect of the store, i.e. how the three (famous) stores affect the price formation. Note that the retail price is set by producers for such category and so retail mark-up does not vary across stores. However, the store might have impact anyway, because other not observable factors such as the number of producers serving the store(and, thus competition), delivery costs, store customer targets might differ across stores. So that's why I am interested in the store effect.

All that said, although I am definitely not an expert in statistics, I know that my dataset is multilevel with Level 3: time observations; Level 2: items; Level 1: stores.
So one basic way I tried (in Stata) is:
1) xtmixed price indep var ||store: || id_item, mle

However, I wonder whether I could use xtreg, re for my analysis. The reason is that I would like to know the effect of each store on price. That is, how Store1, Store2, Store3 impacts on price. So does it make sense to include two dummies, namely store1, store2 (the third store would be used as baseline) in the above xtmixed model?
Alternatively, could I use (after setting xtset id_item)
2) xtreg price indep var store1 store2, re mle?

Would the two models be equivalent (note that I don't have random slope)? After all, doesn't xtmixed add dummies or am I wrong? If not, what are the conceptual differences? And which one should be used?
What if I change 2) in
3) xtreg price indep var store1 store2, re cluster(store) mle ? Does this make sense?

As I mentioned above, the reason I would like to directly include the store dummies among the independent variables is to know their impact, which I guess I can't do using 1). Is there a way I could do it?

Thanks in advance to anyone providing answers to my questions.

Ah...just out of curiosity I stated the following above:
"Suppose initially that items are unique to a store, i.e. the same item is not in two or three stores, but only in one store." What kind of analysis should I do in case the same item can be found in more than one store? Is this multi-dimensional panel? Does stata support it? If so, how?

Thanks again,

Paul