Should Scotland be an independent country?

Well, in the short term there really isn't another credible option*- but could you really see Scotland being happy that the BoE had total control of interest rates and borrowing?

*Unless they start printing and minting Sporrans now

As long as it's doing an okay job, yes. Having an independent central bank is not a bad thing in itself. That's why we have independent central banks. France gets on just fine with their central bank being located in Frankfurt (well... in terms of "face"). At one point Iceland was considering using the Canadian dollar (I don't know how seriously, but it was raised in the media). Of course, having a central bank which is concerned with your particular economy is probably better than having one that's not.

Although I don't really understand this stuff that well, that might be an advantage for all I know- would that mean that the BoE would provide lender of last resort type cover and so forth for Scottish banks?

Not likely if there's not a formal relationship.