That's a lot of questions in one, but here goes.
1) R1,R2 generate Vs/2. If you have a 7.5V supply and Vs is regulated to 15 then you can eliminate them; however Vs/2 is better if Vs can vary.
2) If you got gain around 33 you're doing it about right.
3) Upper cutoff frequency is not determined by any of these components; the datasheet should show it for different gains. Lower : I think you calculated C1/R3. There are two other LF cutoffs : C4/R4 and C7/RL; these look like higher frequencies.
4) For different purposes. Ceramic capacitors are good for low values, electrolytic for high values. And metal film caps are highly accurate and stable, (and some people say they sound better) but I don't see any of them here. Elecrolytics have one big limitation : they are polarised - they should be used with a DC voltage across them, and they must be connected right way round.
5) Generally, no. You can't usually find non-electrolytic caps large enough (though if you can, at the right voltage rating, you can sometimes substitute).
And connecting electrolytic caps in unbiased positions or where voltage might be reversed, is usually a bad idea.
6) no question 6?
7) ah, Q7 makes up for it...
C1 : Coupling. DC block between "In" voltage and pin 1
C2 : Decoupling. Remove noise on Vs/2
C3 : Decoupling. Reduce noise on Vs, smooth out supply voltage
C4 : Filtering. Reduce DC gain to 1.
C5 : Decoupling. Remove HF noise on Vs, improve stability, eliminate HF oscillation.
C6 : Zobel network (good search term). Eliminate HF oscillation.
C7 : Coupling. DC block between "Out" and speaker. Avoids blowing up speaker.
8) Gain = 33 down to DC. (ASSUMING you reconnect R4) Translates e.g. 10mv input offset to 330mv DC offset at output. Not so serious in this configuration because of C7, starting to eat into voltage swing and thus output power. In configurations with +/-7.5V and no C7, applies DC to loudspeaker : bad.
9) If you're lucky, nothing. Otherwise : HF oscillation, horrible whistling noises, distortion, overheating IC, and interference with radio transmissions.
10) Not necessarily. But if that increases the current output, then yes. But in my opinion, C3 and C7 are too small to call this a hi-fi amplifier, 4700uf might be a better choice. (Can you hear the difference? :-)
11) See Q7. But what happens if you omit it?
If you're lucky, nothing - while C3 is fresh and new. But after a few months when it starts to age, possibly: HF oscillation - see Q9.
Best Answer
Simulation shows that this will work just fine - as long as you use it as a preamp, and do not try to drive a speaker. Driving a 10k load it will produce about a gain of 60. Driving an 8-ohm speaker it will have a gain of about 1/8.
So don't try to drive a speaker with it, OK?