Ok, one player. I originally had two and took a page out of Star Wars DarkStryder campaign wherein each player plays three characters, one bridge crew, one mid-rank officer and one security/redshirt. (A suggestion is perhaps Captain (in TOS era, 1st officer might be better in TNG), an engineer, and security.)
Something like that should work for you.
Also. Do the same for yourself and fully develop three special NPCs and play along just as if you were a player too. The trick is to play someone who is equal to and helpful to the player's characters but not better in much so as to not steal spotlight.
Consider also a more cinematic approach to how you tell the episodes. Perhaps try to mimic what the shows do by showing the player what happens even if his character isn't there, for example in a space battle "cut" to a scene where a hull is blown and some unfortunate ensigns are hurtles through a corridor by the balst and so on. The old Star Wars RPG, and TORG were big on this.
As for your father, if he doesn't enjoy it, don't force him. If his willing to endure, try to adress his concerns. Take away the instances that feels like anything can happen. Make it feel more real, and avoid whimsy. No godlike aliens like Q, any temporal or whatever distortion should consistently work in the same manner. Consistency, is the key. Also, there is a huge amount of material available for Star Trek, the movies, the shows. Watcha few with him, and if he enjoys that he might enjoy the RPG based on it.
Apart from Rimmers suggestion there are several things you can do to take away what feels complicated. (I only have the LUG version btw). Do away with specializations, make each character have one flaw each, use the rule for automatic success. Take a narravist approach to space battle instead of the board game rules.
Do story heavy episodes with as little dice rolling as needed.
Then, over time you can increase the complexity, little by little, when you're certain he knows the rules.
The adventure seeds. I simply went through them all and selected those that fit my campaign (a regular ship bound exploration campaign for the most part) and put together a document for them.
The selection process went like this:
Did I get hokked by the description?
If yes, it's in the document.
Will my players enjoy it?
If yes, it stays in the document.
It's really that simple. Since they're just seeds you will still have to do a lot of work, in how your player's characters will fit in, and the particulars.
Most of the seeds are good, but some are more geared to a certain era and won't fit for that reason, or to a certain type of campaign and won't fit for that reason.
Daniel "Warduke" Schenström