How to determine values of M, N, C, charge pump and loop filter for PLL reconfig?

It's a bit involved and sadly, the short answer is on page 5-26:

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The Quartus II software automatically chooses the appropriate scaling factors according to the input frequency, multiplication, and division values entered into the ALTPLL megafunction.

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Going through the same process in the past, I was not able to find an appnote or source code that approximates the behavior of the MegaWizard GUI.

You basically need to write your own software that solves for (M and N) to achieve a stable VCO frequency that is the least common multiple of your desired Cx divider output frequencies.

In my applications, I have relied on passing values precomputed on the PC, and have limited in-system computation of the values to read-modify-write of the Cx dividers to perform simple adjustments (e.g. half/quarter rate modes of operation).

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Is there a unique set of values to get a specific output frequency-phase relationship to an input frequency or can multiple values of this 144 bit (made up of 9 fields in total) word achieve the same result?

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Yes, you can run the VCO at higher frequencies than necessary and use correspondingly larger Cx dividers to get the same output frequency. There may be multiple values that satisfy your requirements.