By aluminum butoxide, could you be more specific? Do you have an exact example you are thinking of, or are you referring to one or more of the examples in the post.

Generally NaBH4 is not a strong enough nucleophile to react with epoxides. You need a counter-ion that is more strongly coordinating to oxygen. LiAlH4 will work.

Generally not. Even though it would release ring strain, you’d be going from a very weak base (Br-) to a much stronger base (O- ) if bromide ion opened the epoxide. Furthermore, even if it did, the reverse reaction would easily occur – recall that epoxides are formed from the closure of deprotonated halohydrins!
https://www.masterorganicchemistry.com/2015/01/26/epoxides-the-outlier-of-the-ether-family/
You really need the presence of an acid to stop the reaction at the halohydrin stage.

Is LiAlH4 even compatible with AlCl3 ? I’ve never seen those two reagents used together.

If it’s useful to you – great!

That’s not really the opening of an epoxide with a base. There, you’re trying to coordinate the epoxide oxygen to the electrophilic phosphorus, forming a transient ion pair that will release chloride ion and open the epoxide. Kind of beyond the scope of what we deal with here.

I’m not certain how to answer your question without a diagram. If you’re looking for selectivity between opening two epoxides, that will be difficult to achieve with NaOH unless there is a clear steric or electronic preference favoring one attack versus the other.