Why are propellers de-iced before the engines are started, and does ice on the fuselage affect the flight characteristics of the plane?
The props are done before starting because you need to make sure the blades and spinners are fully cleaned off while they are stationary. Otherwise, they'd vibrate like hell when starting and shed bits of ice all over. Also, if you just sprayed the props while running the engines would ingest a lot of glycol, which, if it doesn't make the engine flame out, can gum things up and cause cabin smells as cooked fluid makes its way into the air conditioning system through the compressor bleed system (a lot of turn-backs from "cabin smoke" reports on departure are from the stink from cooked glycol getting into the cabin via the bleeds).  

Even done stationary, some fluid will make its way into the engine and it's typical to run the engines at a moderate power setting after the treatment for a couple minutes to try to purge whatever glycol residue is in the compressor before opening the bleeds.

Fuselages are deiced depending on the airplane and the conditions.  On some jets, especially those with tail mounted engines, the upper fuselage is treated as if it was a flying surface and must be clear of ice and frost, even small amounts (it varies with the regulatory jurisdiction).  Otherwise it will depend on whether there is any significant accumulation, mostly for weight considerations.  

If there is an obvious layer of snow/ice on the fuselage, it will be removed... mostly; don't expect a perfect job and they won't normally shoot down as far as the windows. If it's just morning frost, the fuselage may be left alone although the wings and tail must still be done (ANY frost on leading edges can be fatal).  The Capt is in communication with the deicing personnel working the boom and will give instructions on what is to be covered, and what to avoid (like APU air inlet in the tail, or heat exchanger inlets, etc.), whether to use Type I only or Type I and II/III/IV (Type one is for melting/removal and II/III/IV is for follow-on residual protection for a limited time).  

A deice with follow on residual anti-ice protection costs the airline a couple thousand bucks per flight so avoiding unnecessary fluid application is important in the long run.