Abilities
Since you are using Strength, Constitution, Intelligence, and Charisma, all directly equivalent to LL abilities, I would recommend calling Agility Dexterity as well; then Psionic Potential is obviously the correlate to Wisdom, and conversion to LL is straightforward. Changing the name of just one ability to Agility feels "off". (I know how 1st edition S&S did it, but most people who buy this game will not).
Classes and Subclasses
It is not apparent on first read why some of the sub-classes get secondary skills, and others do not. For example, on casual review it seems like Security is simply an inferior sub-class to Fire Control, because Fire Control gets a secondary skill AND gets a +2 bonus in ship-to-ship combat. It might be worthwhile to more explicitly (mechanically) point out the advantages of Security. For example, you might list the special weapons that Security gets access to, or say something like "While Security does not receive the leadership training of Command or the gunnery training of Fire Control, it is the only sub-class to receive training in military-grade weapons such as the ___ and ___. See pg. XX for a full list of equipment available exclusively to Security." The same is true for Engineering, which looks weaker than Communications and Navigation, since it lacks a secondary skill.
Finally, I think it's worth saying what the mechanical bonus actually is in the class description. For example, where you say "Navigation officers get a bonus for detecting space hazards..." say what the bonus is, with a page link to those rules. Even if the player doesn't know what the bonus *means*, it at least will let him write it down on his character sheet for later reference. As it stands, a new player has to jot down a vague note, then flip through the rules to find out what his character's special ability actually is.
Equipment
One problem we've encountered in our S&S playtests is whether such-and-such piece of equipment exists on the ship at all. For instance, let's say an away mission calls for ECM jamming of a Zangid base. The PCs are the command crew of a Cruiser. None of the PCs may have personally spent 3 units to buy an Electronic Interference Device. But has *anyone* on the crew got an Electronics Interference Device? The rules don't say. This is a big problem, because it leaves the PCs unable to really tap into the resources of their vessel. Are we to simply assume that a specialist can randomly be found with any particular esoteric combination of equipment? Maybe this explains why the red shirts always tag along and die - it's the only way to get the right gear for a mission?
I'd suggest either (a) a table with general guidelines for how many of each item might be available in the hands of NPC crew of different ship types; or (b) a random chart for rolling what gear is available from NPCs; or (c) some sort of floating "Ship's Locker" worth of units that the PCs can spend to allocate NPCs.
Psionic Powers
I think the highly random way in which psionic ability is made available to the players could create some unhappy players. S&S doesn't seem to suggest that players will have the same level of casualties as in, e.g., low-level LL, so the PC you roll up is likely the PC you're going to be playing with for a while. As such, having one PC have an X% chance, flat, of simply being better than the other PCs, seems out of kilter. That's how 1st edition D&D handled it, and the Psionics rules were griefed like mad by players.
Here are some ideas for how you might offer psionic abilities:
1. The PC can trade X number of ability score points to become psionically active. The number of points required is based on his race. Andromedans and Taurans pay 0, Rigelians pay 10, Terrans pay 5.
2. The PC suffers an experience point penalty for being psionically active - 15% for Terrans, 30% for Rigelians. Unless your Psionic Potential ability score is high, this penalty will not be worth paying. But a Rigelian who rolls, say, an 18 in PP might be happy to pay 30% XP penalty to take advantage of his aptitude.
3. The chance to have psionic abilities could be based on your PP. Perhaps for humans it 2xPP% and for Rigelians its 1xPP%. That would at least give a fighting chance that a PC who rolled a high PP will get some benefit from it.
4. Following the idea in 3, the PC might get a second chance to unlock his psionic abilities in certain circumstances. For example, interaction with highly powerful beings, training from a psionic, etc.