If you really must avoid the use of the inverter IC, an alternative is to create your own inversion in software, using more of the Arduino pins:
Each of the outputs generated by your Arduino code is being used to generate a positive and a matching negative signal via the inverter, to drive opposing halves of the H-Bridge in the L293.
To achieve the same result in software, if your code uses something like DigitalWrite(pin1, x) on a given pin, you will have to assign another pin and add another DigitalWrite(pin2, !x) immediately following the previous line of code. This will generate the inverted signal, which you would feed in where the inverter output line goes in your current circuit.
If, however, PWM and AnalogWrite() are used as your method of motor control, some slightly more involved coding will be needed to generate PWM that is exactly opposite in phase to the existing PWM outputs. This may well be within your skills (or not), but I'll refrain from coding it for you in this answer.
If you need help with that, it deserves a separate question.
To actually answer the questions:
1) The efficiency is most certainly changed when you use a capacitor so generate a phase shift. The winding's magnetic field is going to be reduced. It will get it turning in the right direction, but not efficiently.
2) Yes, you can (and do) regulate a three phase induction motor's speed by controlling its frequency. They can run (with simple Volts/Hz control) down to nearly zero speed while maintaining decent torque, and up to beyond base speed as well. But, your capacitor phase shift isn't going to like variable speeds unless you use an absolutely huge value capacitor, which is going to be difficult to find since it has to be non-polar.
3) The most common early AC Variable Frequency Drives (VFD) used a totem pole of SCR's to generate a six-step waveform that went to the motor. These had little impact on the actual driven motors windings. The newer drives do need motors with a better rated insulation system since newer drives use a high frequency PWM waveform to generate the low frequency output to the motor. Older insulation systems don't like modern high frequency waveforms.
All that being said, you would be better off with a simple 12V DC motor. If you MUST go AC, and need variable speed, build at least a full three phase inverter, even if only using 6-step technology.
Best Answer
It's not. Fortunately.
While motors present some challenges compared to eg purely resistive loads, they are not especially difficult to drive and it is common to use inverters for this purpose where the use is warranted. The main application area is the variable speed operation of conventional mains operated induction motors - which are usually fixed speed devices when conventionally driven.
Many AC motor driving inverters are available - either
from AC mains - to DC bus - to AC out, or
from low voltage DC - to HV DC - to AC out (less common).
The main target is 3 phase induction motors as these are industry standard, low cost per power out compared to most alternatives and make good use of existing power supply infrastructure.
Single phase versions exist (I have several) and many (but not all) 3 phase drives can be used for single phase motors. A 1 phase motor still needs a 2 phase drive as both leads need to be able to be driven above and below the sine wave midpoint (or a bipolar supply with high and low side switches is needed - which is essentially the same requirement).
So, unless 1 phase motor is an essential requirement a 1 phase inverter is generally less attractive than providing a 3 phase inverter and motor.
3 phase motors "work properly" as induction motors whereas 1 phase induction motors are a compromise as there is no 'true' rotating magnetic field to follow and this must be provided by the motor design.
A block diagram of a typical circuit is shown below.
Input at left is here from fixed frequency mains AC, but could be DC or AC from an inverter. A suitable "high" voltage bus is formed at a voltage somewhat above the peak AC voltage to be supplied. About eg > 330 VDC for 230 VAC out and > 400 VDC for 3 phase 230 VAC. 3 pairs of electronic switches rebuild a PWM waveform which can be filtered to give (here 3 phase) sinewave at variable frequency.
Motor connections shown for 3-phase (star connected) motor and single phase motor. Bus voltage can be lower for 1 phase and for delta-connected 3-phase than for star-connected 3-phase but does not have to be.
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Here is how PWM can be used to make one AC phase.
The waveform at top of image shows a PWM waveform chopping a high voltage supply. Here bipolar supplies are shown with negative OR positive PWM relative to center ground BUT unipolar rail to rail PWM can be used to provide any voltage from one rail to the other.
The lower waveform shows the resultant poorly filtered sinewave.