Does I/O pin’s “pad logic” do debouncing?

I think the term 'Pad Logic' is just a design element.
The logic inside teh dashed lines is present on every other MSP too but not specifically grouped. There are no special elements inside this block that aren't on other MSPs.

ABout debouncing, well, the schmitt triggers do some debouncing. However, this is limited to bounce frequencies above the specified input frequency. For interrupt timing, the minimum pulse width is 20nS, so my guess is that only frequencies above 25MHz are filtered (debounced).

Anything below would limit the maximum input frequency - a debounce circuit doesn't know whether an incoming signal is a bounce or intended. Only your application code can know. There is no universal definition of what a bounce is, other than being an unwanted signal. What's wanted and what's unwanted depends on the application.
Imagine the port logig would 'debounce' an incoming SPI datastream with 16MHz Baudrate. It would make the transfer impossible. But for a pushbutton, the bouncing is usually in the kHz range. So what to 'debounce' and what not? Only the code can know.

However, the main purpose of a schmitt trigger is that it won't 'flutter' if the input signal is near the transition voltge and a bit noisy. It ensures that transisitons are clean unless the 'noise' during transition is larger than the hysteresis. E.g. when the input crosses the upper threshold of, say, 2V, the output goes high. But it won't go low when the input falls below 2V, it msu tfall below 1.5V to make the output fall back to low. So unless the signal is on 1.75V and you have more than +-0.25V noise, the output won't flutter. That's what Schmitt triggers are designed for.

Matt Shira said:

And what frequency is “Pad logic”’s signal edge detection working at, or what is its clock source

As far as I know, debouncing is done in software. It is a task that doesn't need huge resources, nor need (relatively) fast response, so if you already have a microcontroller, I think the most sensible thing to do is to debounce in SW

Matt Shira said:

are pushbutton on MSP430 boards typically debounced by software as you have described, or there is convenient and small-footrpint external circuit to do that

We used this compo for S0 pulses. an at least 20ms low pulse followed by at least 20ms of idle time before the next pusle. We had a lot relay contact bouncing on these lines and 1k/100nF suppressed them all.
However, on later devices we too switched to software debounce (actually scanning the input pins every ms and counting the high and low times to detect valid pulses)

If you already have a tiemr ISR that continuously counts ms or does other things, you may jus tcheck the port state in it and reset a counter if the pin is high or increment the counter (until your threshold limit) if it is low. This waay, when the shreshold is met, the button was pushed wihtout any bouncing for the appropriate time and you can trigger an action. No issues with port interrupts, debouncing timer etc. However, if you don't have a continuous tiemr ISR and want to sleep as much as possible in LPM, then this method is less effective.