Electronic – How should I select the crystal frequency

Most likely the 22pF values used by TI are a compromise (cost / availability). The crystal can generally tolerate a few pF plus or minus the calculated value. I would guess that some empirical testing went into the decision to use 22pF instead of a closer value, or perhaps 22pF was already on the BOM.

Ultimately, even a calculation like what's in the datasheet is based on stray capacitance 'guesstimation'. You have to test whatever capacitor value you come up with and make sure that it works in your end product.

Also, page 20 of the C1111 datasheet that you linked to says 12-18pF is the range to use for the 32.768kHz crystal. Your mileage may vary.

The most important thing to keep in mind is that the capacitor should be tight tolerance with an appropriate dielectric material (one that isn't highly temperature dependent, such as NP0/C0G).

Further reading: here's a link to a good explanation of the topic of how crystals and capacitors interact.

Usually 32768Hz quartz crystals are used in the clock applications. This standard is very widespread and these crystals are cheap and highly easy to find.

In order to switch the clock to work twice as fast, you will need 65536Hz crystal, that is not standard, not so common and probably not so easy to be purchased. For example my local distributor has 32768Hz crystals on $0.1 each and the next frequency available is 1.8432MHz

Another possible issue is the work of the internal oscillator with the changed crystal. It will probably work, but some capacitors might need to be replaced with other values.

And at the end, the last possible problem is the stepper motor of the clock. Is it capable of stepping twice as fast as usual?