Both Thorin Oakenshield and Ori's abilities state "If you control at least 5 Dwarf characters…". Seeing as they are themselves Dwarf characters, do they count towards that total?

In other words, if you had both Thorin Oakenshield and Ori under your control, would you then only need 3 additional Dwarf characters to benefit from their abilities?

Yep, they themselves are Dwarf characters. If you run both with another Dwarven hero (say, Dain), you could bring Thorin and Ori "online" by playing two Dwarf allies.