Hello,
#1 - yes. Kaltura CE is meant to be installed on your own dedicated server[s]. It will allow you to manage you media library from a central location, transcode your source videos so they are optimised for various devices [desktop/laptops, tablet, cellphones, etc] and serve them while taking into account the network bandwidth. It will also allow you to customise the player, add your own social plugins and watermark and much more. You can then embed the player into various external sites.
#2 - yes. You can, provided all the sites you embed the Kaltura player on have access to the API server, no external internet access is needed.
#3 - yes.
#4 - yes.