Business jargon and style.
Company profits are down from last year's figures.
Nouns can be used as adjectives. Company profits here means, the profits of the company: The profits the company made this year.
It is accepted business lingo to use company as an adjective: company policy, company ethics, company risks, company practices. All those mean: of the company.
It is plural because in business, profits, earnings and income are used that way. The first two are countable and the third is not.
That said, on a profit & loss statement (also known as: an income statement, the accounts showing how much a company makes in a year), the singular is used as a line item. The sentence could also be written as: Company profit is down etc. Both would be acceptable. It depends on context.
The determiner "the" is often not used because in business lingo, it's a general statement about a company that has already been mentioned. If your document or context makes clear which company you are discussing, there is no need for "the".
ABC Company [the paragraph is kind of stupid, but it illustrates my points]
ABC Company made a profit [line item idea] last year. This year, the company did not make a profit. Last year, the profit [line item idea] was very high. And company executives have stated they expect to ABC company bring in big profits [earnings or income] next year as sales of blue widgets are expected to grow significantly.
Notice the shift from "a" to "the" in the second mention there. Also, when you make statements in general with a countable noun, you don't use "the": I like an apple for breakfast. I like apples for breakfast. So, too, "company profits".
Those sentences copied from the dictionary are not part of a written text. They are just "unattached" sentences. So, it can be a bit tricky to justify usage.
That said, "The company's income is [whatever]" implies that, in a scenario with the, you are specifically not talking about some other company "thing" such as "The company's outstanding loans", for example. In a press release or memo, you could use: Company income (or earnings or profits) rose by 10% year on year.
[I think I've covered the questions; I can always come back and edit this.]