I think the meaning is close to your first suggested interpretation. I would word it as:
Now we are about to hear the music we have chosen for today.
The two-word phrase here comes is somewhat idiomatic, meaning "something or someone is arriving at or approaching the place where we all are now." There's a song often played at weddings that begins with "Here comes the bride..." which means, essentially, "The bride is here!" Similarly, "Here comes the bus" implies the speaker can see (or hear) the bus off in the distance. So, when the speaker says, "Here comes the music today," I interpret that as, "Today's music is about to be heard."
It might be worth noting that the last paragraph isn't very well written. For the most part, "music" is an uncountable noun, so I'd offer this suggested edit:
I shall explain why they are all good songs... (or compositions, perhaps)
or:
I shall explain why it is all good music...
To me, studying implies that you are learning something new. It's usually used in an academic context:
Sorry I can't go out; I need to study for a test next week.
Bill is going to Oxford next year where he will study history.
The zoologist spent 8 years studying insects before she wrote that paper.
but it can also be used when conducting some sort of analysis:
We will be studying the safety habits of our factory workers to see if we can't cut down on injuries next year.
Work can imply physical work, as you mention, but it can also be used when describing any kind of mental exertion:
It took a lot of work to solve last week's crossword puzzle!
I'll be working on my term paper, so I can't go out with you guys tonight.
I'm working on coming up with some fun ideas for the baby shower.
In short, work is a very general term, which study seems more narrowly focused.
As to Jamie's quote:
I have to finish and hand in my project by Monday, so I have to study/work for the next few days.
I think work is the best word if Jamie pretty much knows what to do already, and it's just a matter of doing the work. For example, if Jamie is writing a computer program, or a term paper, or painting something for an art class, that's generally work.
However, if the project is a matter of answering 10 short answer questions, but Jamie will need to do a lot of research in order to answer those questions, then the word study might be appropriate.
Best Answer
"Shall" and "will" mean basically the same thing -- they are referring to the future. The traditional grammar rule is that "shall" is only used with the first person ("I" or "we"), and "will" is used in all other cases -- but nobody actually follows this "rule" with any consistency. Effectively, "shall" and "will" are interchangeable, except "shall" sounds a little more formal.
(Here's a Wikipedia page about "shall" vs. "will".)
"What do we do now" is in the present tense, and is the most correct of the three options, since the speaker is talking about actions in the present. "Will" and "shall" are future tense -- so "will" is correct if you are asking, "What will we do after the movie?"