As to your questions, I don't see any reason why not, and yes.
The issue with fighter-sized intersystem drives was power (fighters didn't have enough), and it wasn't until the end of FS1 that they invented one that was viable. But cruisers had intersystem drives, and I don't see why any docked fighters wouldn't be pulled along with it.
As for the freighters/transports, we saw them jumping through nodes all the time in FS1. Yes, they had intersystem drives.
I agree with Kie99 about the communications thing. I think all that was required was some sort of relay in each system you wanted to transmit through (Beta Aquilae communications station). Whether anyone was blocking the node at the time is irrelevant.
What Kie99 said about intrasystem jumps inside nodes needs a bit of clarifying, though: ships seem to be able to intrasystem jump away from a node. The only reason they fly away first is probably, like he said, due to recharging jump drives. However: every single time a ship has needed to jump to a node, they had to fly a few klicks in normalspace beforehand. The recharging drives thing doesn't work here because one thing we do know about intrasystem jumps is that they consume very little power... even a fighter can manage them effortlessly. It's the intersystem jumps that eat a lot of power. Notably, the one time a ship attempted to intra-jump into a node (well, a Knossos, but same thing for our purposes), they got flung 9 klicks off-course. This suggests to me that there is some kind of disturbance of subspace near nodes, and intrasystem jumps within a few klicks of the node are not possible coming in. It's true there's no documentation on this anywhere in the games, but all evidence points to it. Since these few klicks of flying to a node leave them very vulnerable to attack, there is absolutely no reason for them to do this if they could just jump into the node from across the system.