Effect on current draw and voltage when splitting a signal to many places
What kind of voltage drop (if any) am I likely to experience per split 


That depends on the input impedance of each individual input, and the output impedance of the output. 

If the output impedance is low and all the input impedances are high, you will not see significant voltage drop.


  (or more importantly, how many splits could I get away with before the voltage dropped significantly).


The lower the output impedance, the more splits you can get away with. The higher the combined input impedances, the more splits you can get away with. 


  Will each extra input that I send the signal to cause extra current to flow through the output, or will the current just split between each input?


Yes and yes. Strike the "or".


  Are there any other things I'll need to consider to avoid any catastrophes?


Don't draw too much current, in other words make sure the input impedances are high enough for the output. 

Google Ohm's law and Kirchhoff's laws. Memorize the former. Understand the latter.

Get acquainted with electronics and safety. Electricity can kill or start fires, so make sure to inform yourself, so that you're able to make judgements on whether a certain circuit is potentially dangerous or not, how to stay safe, and how to design stuff that is safe.