Why do Tesla Coils shoot off electrical sparks?
They generate extraordinary voltage by using a series of step up transformers. Step down transformers are like what's in your computer, taking mains down to 5 volts. These will happily work at 50Hz frequency that your mains comes in at. But step up necessitates using higher frequency alternating current. This is due to the induction equation...
Voltage Induced = -L * Rate of change in Current.
(where L is a constant.)
The faster you change current, (higher frequency) the bigger the right hand side, so the bigger the induced voltage is.
Tesla coils produce these very fast but short lived lightning bolts, that you perceive as the sparks.
Would a Tesla Coil work in the Vacuum of space? would they still emit electrical sparks?
Yes and no. Electricity looks different in space. What we see in the blue colour of nitrogen gas being ionised. In a partial vacuum electricity causes a glow rather than sparks: like with neon tubes. The principle is very similar in space; but you may not get any visible light in a true vacuum.
Since Electrons have a negative charge would a strong 'positive pole' magnetic field cause the electrons being emitted from the Tesla Coil to be deflected/repelled?
I read somewhere that if an Electron is moving and hits a magnetic field it gets deflected.
Yes. A magnetic fields will bend an stream of electrons. This is exactly what happens in a motor in fact. Just that the electrons are clinging to the wire as they flow.
The electrons don't care if they flow with a wire or without, they'll respond to a magnetic field in just the same way.
The electrons feel a force, causing them to turn in a circular arc in a plane at right angles to the magnetic field.