Characteristic impedance vs. Resistance
Why is there voltage drop across resistors?


Because that's the definition of resistance, \$R=\frac UI\$.

If a device doesn't have a voltage drop if current flows through it, it can't be a resistor.


  Characteristic impedance is the ratio between voltage and current in a transmission line, but resistors also have impedance that translates to voltage drop. Why does power dissipate across a resistor? And how can a transmission line be lossless?


Because one is a characteristic impedance, and the other a resistance – you should probably just read on a few pages in your transmission line theory book, and will understand the difference.  They have the same unit, because they're a voltage per current, but they don't describe the same phenomenon.

In a resistor, a charge passing through it loses energy (energy per charge = voltage). In a transmission line, we can't speak of charges flowing (there's no net current of electrons in direction of wave propagation), but have electrical and magnetic Field Strengths, given in Volt per Meter, and Ampere per Meter, respectively. The ratio of these fields give you the wave impedance.