You're looking too much into this, imo. People rave about dual tree build because they offer the maximum amount of flexibility while having both roles of the TB fulfilled: support (ice) and damage (fire).
If you go all ice, you'll have a great deal of supporting capabilities, and your team will love you. However, since you sacrifice nearly all stopping power of your own, you have effectively forced yourself to always travel with a companion or risk feeding the enemy team with your deaths should you try to fly solo for a prolonged amount of time.
If you go all fire, you present top bursting capabilities up until the very late game, and you can become a hunter of nearly all DG's, regardless of your level of support from your team. However, your lack of support options will make you less effective at producing any kind of synergy with teammates in a situation where you're outclassed(i.e. being able to win a 2v3, or buying enough time for you and your teammates to escape). Furthermore, ability-based builds (which is typically all a pure fire TB can use) pale in the face of late game uber auto attack damage and stupidly high HP pools. If the game drags on, you're in trouble.
So how about a hybrid build? Well, the cons are giving up a degree of the pro's listed above in the respective "pure" builds. The pro is having (to a degree) the best of both worlds. Common builds are frost TB's that include maxed fireball (including the lvl 15 synergy ability) which allows TB's to have a super effective opener (or finisher) while still maintaining crowd control and support abilities. While it's less common to see fire-oriented hybrids (I'm excluding people that max shatter for 1350 fireballs), one such build might include frost nova to offer snare capabilities, as well as a potential getaway skill for yourself and teammates.
As far as equipment is concerned, that depends on how you want your TB to behave and what your team and the opponent's team consists of. Personally, I pick my equipment builds entirely independently of whatever tree route I'm taking. However, if anybody has equipment builds that stand out for a particular tree, feel free to correct me here. 
Oh, and to get to the assassin only favor menu, click on the other tab lined up on the left side of your favor item list (same for generals).