Learn Hardcore Japanese: kotosi ha uriage ga tasyou agarimasita.

今年は uses は to mark “this year” as the topic of the sentence (as for this year…). It sets the frame or contrast (implied: compared to last year).
売り上げが uses が to mark “sales” as the subject of the intransitive verb 上がりました (“rose”). が is used for the doer or the thing undergoing the change/event.

What exactly does 多少 mean, and what part of speech is it?

多少 means “to some extent,” “somewhat,” or “a little.” In this sentence it functions as an adverb modifying the verb 上がりました. (Note: 多少 can also act as a noun when you say things like 多少の問題 “some amount of problems.”)

Why is 上がりました used here? What form is that, and why is it intransitive?

上がりました is the past polite form of the intransitive verb 上がる (“to rise,” “to go up”). Because sales “rose” on their own, there’s no direct object. The pattern is: 売り上げ (subject) + が + 上がりました (intransitive verb).

Could I say 売り上げを上げました instead? How would that change the meaning?

Yes. 売り上げを上げました uses the transitive verb 上げる (“to raise/bring up”), which means “we actively increased our sales.” The original 上がりました is neutral/intransitive—“sales went up (by themselves).” The transitive version emphasizes an agent doing something.

Can I replace 多少 with 少し? Are they interchangeable?

Often you can. 少し上がりました and 多少上がりました both mean “it went up a bit.” Nuance: 少し is more straightforward “a little,” while 多少 can feel slightly more formal or vague—“to some degree.”

What does 売り上げ cover in meaning? Is it the same as 売上高 or 売り上げ額?

売り上げ means “sales” or “revenue.”

売上高 is a more formal/business term often seen in reports.
売り上げ額 explicitly adds 額 “amount,” so “the amount of sales.”
All are related but vary in formality and precision.

Could we drop 売り上げが and just say 今年は多少上がりました?

Only if context already makes clear what “rose.” In a standalone sentence, omitting 売り上げが leaves the listener asking “What went up?” You can omit it in conversation after you’ve mentioned sales, but generally you keep it for clarity.

Where should adverbs like 多少 be placed in Japanese sentences?

Adverbs normally come right before the verb (or adjective) they modify. Here 多少 precedes 上がりました. You can also place them earlier in the clause, but directly before the verb is most natural.

How is 売り上げ pronounced (reading)? Any tips on remembering it?

It’s read うりあげ. Break it into two parts: 売り (うり) “selling” + 上げ (あげ) “raising.” So literally “the thing that selling raises,” i.e. “sales.”