You are correct (for the most part).
As you level up you do indeed get to use your bardic performance for 2 additional rounds per day. These come from the bardic performance class features.
However Favored Class Bonuses (FCBs) work a little different than you seem to think in your question. You seem to be implying that you automatically receive your racial FCB when you level up, this is not the case.
Favored Class Bonus:
Each character begins play with a single favored class of his choosing—typically, this is the same class as the one he chooses at 1st level. Whenever a character gains a level in his favored class, he receives either + 1 hit point or + 1 skill rank. The choice of favored class cannot be changed once the character is created, and the choice of gaining a hit point or a skill rank each time a character gains a level (including his first level) cannot be changed once made for a particular level. Prestige classes (see Prestige Classes) can never be a favored class.
And a little further down when discussing how racial FCBs work
Thus, rather than taking an extra hit point or an extra skill rank, players may choose for their characters to gain the benefit listed here.
This means that when you level up in your favored class, you gain the choice of either +1 HP, +1 Skill Rank, or your racial FCB (in this case +1 round of bardic performance).
In this case, if you choose to take this racial FCB, you will indeed get 3 additional rounds of Bardic Performance for your level ups (2 + 1).
Note that as a half-elf you qualify for Elf and Human FCB's as well.
so you can also choose from
Add one spell known from the bard spell list. This spell must be at least one level below the highest spell level the bard can cast.
Add +1 to the elf’s CMD when resisting a disarm or sunder attempt.
when selecting a racial FCB.
Finally, welcome to Pathfinder :D