1. How does a healer go about not stealing threat from the tank?
This really shouldn't be a problem. If it is, it's most likely a tanking issue.
You could try holding back on your heals for a second or two at the start of the fight to let the tank lock down aggro, but it shouldn't be necessary. Basically, if the tank hits a mob even once, you shouldn't be able to pull that mob off him for the rest of the fight.
What should I do if I aggro an enemy that just appeared?
Run to the tank. Stand in front of the tank. Jump up and down in front of the tank. Keep yourself healed up. Don't panic.
Most tanks have a cleave or an AoE in their rotation, so as long as you're standing in front of him, he should eventually grab that mob off you even if he never notices that it's attacking you.
2. Do healing addons really significantly increase your effectiveness?
An add-on won't play for you, it merely presents the information you need in a different way. I use Vuhdo and Weak Auras because I like being able to customise what information my UI displays. They make me a more effective healer. However, if you are happy with the default UI, more power to you!
At low levels they are completely unnecessary. As you get more spells with trickier interactions and/or timings you may find that they become more useful. Of course there is a period of adjustment while you learn how to use an add-on, so it may be easier to set it up earlier on and level with it than to try to relearn how to play at level 90.