That is proper procedure to leave cockpit seacocks open all the time. The seacocks are there primarily to allow the cockpit scuppers to be closed in case of any kind of troble with the hoses, clamps etc. and to allow them to be closed for serviceing of the scuppers , thruhul, or related parts. That said, they should be "worked" closed and opened a few times a month to be sure they will continue to operate easily. This goes for all seacocks, that they should be occasionally worked and not allowed to remain either open or closed month after month, or year after year. You want to be sure that they will operate easily if you ever need to operate them in an emergency.
In reponse to the bilge pump being wired directly to the battery. It isi considered good practice to wire directly to the battery or 24 hour circuit, just be sure to include proper protection for the circuit, such as an inline fuse, or dedicated breaker. Rick