Attacks do not usually have Aspects. Casting a Fireball (or whatever) usually just means rolling your Magic skill (or whatever you're using) and you can tag the Troll's "Vulnerable to Fire" aspect. You could also tag your own "I am good at Fire Magic" aspects to double up on it, though.
It will cost 2 Fate points to do both, but in this case you can indeed get a double bonus. This is because there are, genuinely, two things that emphasize why burning the troll is going to work: you're good with Fire and it's vulnerable to it.
For your second suggestion, if there are the Aspects of "A room full of spiders" and "Arachnaphobia" in play then that doesn't automatically mean you can tag both to get a double bonus.
The reason is that you need to explain why the Aspect you tag is helping you succeed. While your opponent could tag "Arachnaphobia" and point out that there are spiders present that are distracting you, it doesn't work the other way around.
Your opponent cannot point to the presence of the spiders and then say it's making your life harder because of your Arachnaphobia, because "the roomful of spiders" is not the thing making your life harder in this case. Your Arachnaphobia is; but he already tagged that. In order to really tag the "roomful of spiders" aspect, he would have to actually use the room or the spiders themselves; ie by pointing out that you will fail to find the Key of Opening Plot Relevant Doors because the cobwebs are blocking your vision, or that you can't pick the Lock of Relevance because the poisonous spiders will bite you.
The key is to look at the specific Aspect and ask "how is this Aspect influencing the situation?". "Fire based" and "Fire vulnerable" for example are not dependent on one another; you can still fireball a human (who isn't "vulnerable to fire" the way a Troll is), it just won't be as easy to tag anything for a bonus.
Likewise; a non-Arachnaphobic person might still be hindered by a room full of spiders. They won't be as easy to hinder by the mere presence of spiders, and they won't be hindered as much, but the physical presence of a bunch of potentially poisonous, web-spinning creatures might still bother them.
If you want to tag beyond the specific vulnerability, you need to make it clear how even a non-vulnerable person would be hindered and then tag the generic aspect. If you just want to point out the vulnerability, you just spend a single point and point out the vulnerability.