I thought 5 days was fine for the referral program. The lack of available content by contrast wasn't. I think opening up until Starbase 24 is a much better idea as players get to experience more content, though it'll still be short.
I remember having tons of fun during the Open Beta's first week, getting from start to the level 16 cap - which also involved most of the missions in the Klingon Front and content including the Sirius, Regulus, Pi Canis and Eta Eridani sector blocks. When I capped, I still found I could do a lot of stuff I had not experienced, such as PvP - and really, PvP is pretty neat in the Lt.Cmdr tier.
I was thinking that even if STO wasn't going to adopt a free-to-play scheme that it would still perhaps not hurt to apply some of those thoughts to the Demo in order to offer similar content to what I experienced in the first week of the Open Beta. The demo client wouldn't even need to have a time limit imposed to it either: while generous, it would allow people to play through repeatable content such as low-level PvP, deep space encounters, star clusters and fleet actions - these people playing for free would help keep the server population high and give a much more active and welcoming environment to new arrivals (...well, STO players aren't that verbose, but at least it wouldn't be deserted).
People motivated in seeing more content, access to sectors and missions beyond the Klingon Front, more experience, promotions, new ships and such would have ample incentive to go and grab the box/subscription to move on. I mean, it worked for me! Playing the Open Beta and enjoying it was what made me buy the game's box and invest in a lifetime subscription.
btw - we are currently working on ways to reduce the size of the demo download. We nuked several gig off at season 2 and we're hoping that in upcoming months we can get it down even further so that players have less to download in order to try the game.
Champions Online also has been working on this tech too as it will definitely help them when they go F2P.
As for the tutorial, experienced players should be allowed to skip it as an option. But I actually LOVE the storyline and how it ties in with Khitomer Accord STF. Best story progression in the game IMO. The story starts in the tutorial, and ends in Khitomer, one of the end-game STFs.
Also, if you do a new tutorial, could you get a voice actor that sounds like he cares a little bit about it?
Just while we're on the subject, why on Earth don't you approach Robert Picardo if you're going down the EMH route again? I'm sure he'd love to do it. Of course if you're re-working the tutorial completely then it's a moo point. It's like a cow's opinion--it doesn't matter.