Leif - a brief response to a couple of your points :-
do you continually rush after the puck, or place yourself in a good place to intercept the puck,
After spending weeks of puck chasing, I have finally been informed (must be an initiation thing, or just entertaining for seasoned players to watch newbies chasing the puck) that the best bet is to mark a player (preferably from the opposing team), and stick to them. As Gretzky said (and I paraphrase), 'Be where the puck is going, not where it's been'. So I try and mix the two now. My 'game sense' isn't good enough to read where the puck is going, but the idea of shadowing an opponent seems a bit more useful.
during normal public skating you are not allowed to take a puck and stick onto the rink.
I would contend this....if you take a puck and stick onto the ice in a public skate, only positive things will happen. When people come up and complain, you can sneer at them, growl intimidatingly, and skate off. When the rink steward approaches, you can practice body checks, sneer, growl, and skate off, and then when security get onto the ice, you can brush up on your speed skating skills, sneer, growl, and practice being on the end of a body check! Quite a learning curve :-)
I am doing the Swindon LTP session tonight, and I have a nice new Washington Capitals jersey, so I'm bound to skate better as a result!