I will edit this with the appropriate equations and timeframes in a few hours when I get back to my office. For now I will address the foundations of your questions with some basics.
1: The radius of the frost line is determined by the star itself (edit: More precisely the nebula that the star forms from.), more in-depth answer will come with edit.
2: The fact that most of the data for these systems is from methods that favour large planets close to the parent star, means that our selection isn't biased, but the method instrument combination is. Another possibility is that migration and loss of angular momentum could be the culprits in this scenario. It is also worth noting, that in the next decade there should be significant data to state whether our solar system is actually common, an anomaly or just younger than many of the other solar systems observed thus far.
3: Larger gas giants "Jovian" planets like all orbiting objects migrate inwards as they loose angular momentum. During the oligarchic growth phase of a soon to be large planet the forming core accrets a lot of material while clearing "all" of the material within it's orbit and associated jeans radius, as the core migrates inwards the supply of material is once again increased.
In short that is why there are "Gas Giants" within the frost line, things migrate inward as they loose angular momentum, and "Ice Giants" form beyond the line because they took too long to migrate inward and gather enough material to move to the next level. Also, if the protoplanet takes too long to accret a ~10-15 earth-mass core, all of the accretable material will be ejected from the system by the star before any significant atmosphere can be obtained.
The dwarf planet part, has to do with the amount of material and how long the protoplanet took to accret it. In that respect, the ice giants and the dwarf planets share a common shortcoming.
I find myself overwhelmed at work and unable to fully explain the entireties of the process at this time, please forgive me.
However, I do offer a link to an upper division astronomy class' notes that gives a very good explanation of what I believe will help you with your question.