"Part" can be countable or uncountable. If you mean it's one of several parts, you could say "a part." But in your example, "part" is being used as an uncountable noun since the other "parts" are irrelevant.

Not 100% sure but doing the best I can: "a" is often omitted, and perhaps best omitted, in "[a] part of [X]" where [X] is or stands for a singular noun or uncountable noun (in colloquial usage perhaps also for plural countable nouns), and "part" means something like "share" or "proportion" or "aspect" e.g.: "part of what I like about her is her sense of humour; I'll swap you part of my lunch for part of your cake; the problem is complicated - part of it is his attitude, part of it is his lack of talent; I'll eat part of the cheese for my breakfast." But not "I'll swap you some chocolate for part of your marbles" ("marbles" being plural / countable), at least in careful standard English, though you may hear it from some native speakers. In all these examples "part of" is a fairly vague concept of a share or proportion or (in your example) an aspect of a broader/larger thing, and does not refer to a specific, physically separate component of a separate/distinct nature. "He wants to sell part of his land" is fine, though the part in question may be easily defined and identified. "He needs to buy a part for the car" refers to a specific component, e.g. a gear-box, and needs "a".

That's a good question, it must be confusing for English learners. To me they are the same. I cannot think of an example where the two phrases have different meanings.

Hi