The stats required for assassins and sorcs on the armor are the same (both want willpower and endurance) as are the class of armor (both light), so there is a fair amount of overlap between them. DPS assasins and sorcs will want the same/very similar armor. Healers are pretty close to the same. Tankish assassins will have some differing optimal stats (more endurance and defensive focused secondary stats). Since most of the stats are depending on swappable mods, the appearance for all of them can be the same.

Technically, no you can do them anytime. However, if they are grey on your mission tracker you will receive no xp for them.

Basically get to the station just off your starter planet. There you can trian in the crafting/gathering skills by simply talking to the various trainers and selecting "yes" (or some variant of "yes") You get a maximum of 1 true crafting profession/skill (biochem, cybertech, armortech, armstech, artifice, synthweaving) and you pick up to 3 of the mission skills (investigation, diplomacy, underworld trading, treasure hunting) and gathering skills. You are however, limited to 3 skills total that are shared among your entire crew. Each crafting class has a gathering skill and one or more mission skills that go well with it. You will get a codex entry after talking to each trainer giving more details on what is the natural pairing for each crafting class.

It shouldn't be too bad. You get all the same abilities that lightning spec gets until L20 IIRC. You miss a few little buffs, but it's a few percent here a few percent there. The drop off in damage isn't going to be very large, but it will become more noticeable after the lightning spec gets chain lightning and you move into higher levels. I'd be sure to run Khem in DPS mode for the added damage if you find it at all slower than you'd like. While you're built for healing, you can and should take a fairly aggressive approach and deal out damage in most solo encounters. Proactively use the bubble shield (I forget the name offhand) to minimize the time spent healing, it's one of the best powers available.