Potentially yes you can keep it running in background. But in general terms, the longer it runs the better it will be.

If the connection was fine from the beta, use that connection cache.

I just connected with BS Beta on Win 2000 & as usual it connected quickly. Snapshot below (I edited out ip addresses & host names for privacy.)
The no. beside the earth symbol I'm not even sure what it means. However the number to the left of it shows how many peers you are connected to. Mine shows 16, & the Host connection window verifies this as you can see below.



I'm often connected to more polish peers on Windows 2000, however it may depend on time of day.

I had not updated the cache I posted in the connection thread since last year. I've just posted a fresh choice a minute ago.