"open/close" SqlConnection or keep open?

Stick to option a.

The connection pooling is your friend.

Always close connections as soon as you are done with them, so they underlying database connection can go back into the pool and be available for other callers. Connection pooling is pretty well optimised, so there's no noticeable penalty for doing so. The advice is basically the same as for transactions - keep them short and close when you're done.

It gets more complicated if you're running into MSDTC issues by using a single transaction around code that uses multiple connections, in which case you actually do have to share the connection object and only close it once the transaction is done with.

However you're doing things by hand here, so you might want to investigate tools that manage connections for you, like DataSets, Linq to SQL, Entity Framework or NHibernate.

Use Method (a), every time. When you start scaling your application, the logic that deals with the state will become a real pain if you do not.

Connection pooling does what it says on the tin. Just think of what happens when the application scales, and how hard would it be to manually manage the connection open/close state. The connection pool does a fine job of automatically handling this. If you're worried about performance think about some sort of memory cache mechanism so that nothing gets blocked.