The use of past perfect is the backshifting of the action taken before the time which the fragment is describing. It's backshifting of present perfect tense. I'll explain.
Philip was under Hayward's influence prior to the Christmas Day. Under that influence, eventually, also prior to the Christmas Day, Philip's opinion (that the festivities were vulgar) formed. When, hard to say exactly. What is known, however, is that on the day of which Maugham writes the opinion was already there.
To express the action that took effect some time in the past, at an unknown moment, but of which we have a definite result, we use present perfect. If we go back to that day, we can say that Philip has persuaded himself because the result (the opinion about the festivities) is apparent, but the moment of its formation is unknown.
If we take the description entirely into present tense, we have:
The result is that Philip has nowhere to go, and he spends Christmas Day in his lodgings. Under Hayward's influence he has persuaded himself ...
Now, push everything into past tense and you should turn "has" into "had" (twice):
The result was that Philip had1 nowhere to go, and he spent Christmas Day in his lodgings. Under Hayward's influence he had2 persuaded himself ...
Now, about why not both in the same tense (your second question).
First off, it's author's prerogative to use whatever tense seems suitable to them. Second, if we treat the text of the book like we do Nature, we can't say what something ought to be. We can only attempt to explain given the evidence.
The evidence in this case is that "made up his mind" is in past indefinite tense, and not in past perfect. What does that tell us? I see it that it tells us that making up Philip's mind most likely coincided with other events described.
... festivities were vulgar..., and he made up his mind...
To put it in present tense (as if we are there living right alongside Philip)
... festivities are vulgar..., and he makes up his mind (now) that he will take no notice...
That is, he didn't make up his mind before that time of which the author speaks to us. Which time is that? Some time soon before the Christmas day, probably, since there is the "will take". Right after that, Maugham writes "but when it came...", pointing out that the actual day followed.
In other words, we have three events, Philip persuades himself, Philip makes up his mind, Christmas day comes. The moment of persuasion is uncertain, but in the past relative to the moment of making up the mind. Both are in the past relative to the Christmas day.