The cloaking device shows a speech bubble with an on/off button, click that to start the cloaking cycle. You stay cloaked for the duration the upgrade screen tells you, after that it's a 20 second cooldown (each blue circle on a stunned system corresponds to five seconds).
Boarding is awesome but I don't think I can give you too much advice since my tactics all revolve around the crystal lockdown ability. I did notice that the AI does not redirect guys to your boarding force until you actually damage the room you're in, if you teleport on top of a guy the AI will often not reinforce until he's dead. Avoid fighting in enemy medbays though, the regeneration is very strong. Disable that thing with your ship weapons if you have to (preferably just after badly wounded enemies ran into it to get healed). Oh and if you have reason to believe that the enemy will jump away then get your crew outta there, any crew left on their ship when it FTLs away is lost. I don't know how advisable it is to board automated ships with normal crew, crystalmen can survive long enough to break a subsystem and teleport out again but you usually don't get them.
I wouldn't aim for the engine first unless you get a specific message telling you that the enemy tries to escape or you need to lower his evasion. Disable the weapons as soon as you can, no weapons = no damage. Unless he's got some nasty drones, then you should shoot the drone system. On later enemies you won't get around shooting the shield generator so do that until the shields are low enough to manage, then go for the weapons. Also note that whenever the enemy re-routes weapon power to other guns (they slide in/out) their reload bars are reset so they won't fire soon even if they do get repaired quickly.
I try to stay in a sector until exactly one jump before the rebels reach the jump beacon. There's no value in engaging the rebels, the game will not give you any real rewards for beating the advanced fighter you face when you get caught (presumably to prevent people from grinding the rebels). Don't put too much emphasis on getting many nodes though, the rewards scale with the difficulty of the sector so the loss from visiting fewer nodes isn't that big, it's more important not to waste your resources on the rebels.
EDIT: The teleporter has two buttons (one for teleporting out, one for getting people back), it'll teleport everybody in the teleporter room when outgoing (even if it has more people than fit into the destination room, it'll place the surplus in a nearby room) and everybody in the target room when recalling.