What is the synchronous speed of single phase induction machine from 2 phases of 3 phase machine in parallel?
Yes. You can do something like that. In order to get the motor to start, you will need to connect a capacitor in parallel or in series with one of the windings. The number of poles and the synchronous speed will not change. The starting torque will be reduced. The slip and associated losses will increase. The safe operating load will be reduced. There are more complicated connections using multiple capacitors that can improve the performance to some extent. One such arrangement is called a "Smith Connection."

Since this use of a three-phase motors is not what they are designed for and since motors that are designed for single-phase operation are better suited for the purpose, there is very little detailed information available about motor performance using these types of arrangements.

Here are some diagrams and additional information. Reference Link

Edit: When I said "losses will increase," I should have said that the consequence of increased losses is reduced efficiency.

I assumed that you were asking about operation as a motor. The motor will also operate in the zero sequence braking mode with a parallel winding configuration with the same synchronous speed as for the series connection. You would need to apply a lower voltage. Switching from motor to brake operation would be more complicated. The easiest implementation involves a motor that is delta connected for motor operation with the delta opened and voltage applied to the two opened ends for braking.

For a wye connected motor and a parallel connection for zero sequence braking, I assume that you would disconnect from power, connect the three motor phase terminals together and apply voltage between the phase terminals and neutral.

Edit 2: Since you are only interested in braking, you may have some kind of continuous braking application in mind. Zero sequence braking will return some of the braking energy to the supply and thus reduce rotor heating, but the percentage of the baking energy that is returned will apparently be rather small. In addition, the braking motor will be be injecting harmonic current into the power system. Keep in mind that the applied voltage must be lower that the normal winding voltage to avoid overheating the motor. Connecting the windings of a delta connected motor in series reduces the winding voltage to 1/3. Since that scheme is apparently practicable, the motors must be able to tolerate that for occasional braking operation. For continuous braking, further voltage reduction may be required.

Reference