Weren't you suppose to discuss with your legal department about trademark issues with using the Satoshidice name? Satoshidice was pretty much the very first dice site ever created on Bitcoin network and it was very successful and sold for a huge sum back when Erik sold it. Its a piece of pioneer work in the bitcoin history.
Hence by reusing the name you are creating confusion because people might watch that Netflix bitcoin documentary which mentions Satoshidice, then go Google it and find your site and think its the same. I see some legal issues with this.
Yes we are thinking about that also. Once the site is stable in terms of user traffic, we might consider switching to another name, as changing name and domain would require additional fundings to be done.
That would not be a right strategy. Either change the domain name before starting the casino business or stick with the same name. Once people are used to playing at Satoshidice and your site would be popular, it will not be a good idea to change the domain name at that time. I suggest you stick to this name.