With the exception of power op-amps, op-amps are not suitable for driving loudspeakers. The impedance of speakers is too low. You need a power amplifier chip for driving speakers, or to add a class B emitter-follower output stage made using two complementary power transistors.
Some op-amps are much more noisy than others. They produce noise and it cannot be eliminated; it is inherent to the op-amp. If you're doing audio work, and the circuitry doesn't have to be very low power, use an audiophile op-amp IC like LM4562 or the cheaper workhorse that is nearly as good: NE5532.
But neither of those will drive an 8 ohm speaker, or even a 32 ohm headphone speaker.
Also, do not expect to get anywhere near the 0V and 9V power rails with the signal swing. It's not realistic. Even if you get an op-amp that can do it, the speaker-driving output stage will not. I wouldn't plan on more than a +/- 2V displacement from the 4.5V middle bias, staying several volts clear of either power rail.
I sketched up the circuit below to illustrate the class B emitter follower output stage idea. The op-amp is basically used as a buffer with a modest voltage gain of 2 to interface the input to the output stage. You don't need a high gain because the available voltage swing is not so high. The output stage will only swing the voltage on the speaker to about +/- 2V before clipping. If the input signal is already +/- 1V, the gain headroom is only 2!
Note how everything is very similar to your original schematic, except that we have added the output stage, connected the feedback resistor R2 not from the op-amp output, but all the way from the output stage. This includes the output stage in the feedback loop, and so the feedback loop will cancel some of the distortion introduced by the output stage, such as its crossover distortion.
We still have serious problem here in that the output stage's input impedance is still too low for the op-amp! It's not nearly as bad as an 8 ohm speaker, but look at those 470 ohm resistors. Basically, they have to be considered in parallel, and so the input impedance is only about 230 ohms! Their values can be raised, but this cuts into our already limited voltage swing. We can fix that by using pairs of transistors in the output stage, to increase its current gain.
I can think of an op-amp off the top of my head which can handle this impedance as it is: the NJM4556. The op-amps on this chip can drive 150 ohm loads.
simulate this circuit – Schematic created using CircuitLab