LM7805 followed by LD1117-3.3
Yes, no problem. 1000uF is not much bigger physically. 
If they are close to each other, perhaps. Put it closer to the LM1117. 
Partly history of the applications, but the LM1117 actually needs a capacitor on 
the output with certain characteristics to keep it from oscillating like a    banshee.
With the LM7805 the caps are not generally
necessary- they just improve performance (lower output impedance, improve transient response). 
They can all be ceramics (including the 10uF), however you would need to put a series resistor on the
10uF in accordance with datasheet ESR limits. It's probably worth the extra part to avoid using a tantalum. 
Most likely the 1000uF is also a filter cap. Going higher in value is 
usually better with capacitors and those are really common values for an off-mains
supply for moderate current. 47uF/10V or 100uF/10V is not much physically larger
for the output  cap and I would tend to use that. 


Edit: Regarding the output capacitor requirements of the LM1117, below is the information from the TI (nee National Semiconductor, the original designers) datasheet for their LM1117: 



So something like a series 0.5 ohm resistor should be fine. Note that a 10uF ceramic capacitor has a (negative) voltage coefficent as well as a tolerance, so you may wish to use a higher voltage rated part or a higher nominal capacitance part to ensure the 10uF minimum requirement is met, just to be completely safe.