given that SpaceX is choosing what price they're charging starlink, there's a reasonable argument that starlink isn't profitable either

That's in the IPO documents. Starlink had $11.4 billion in revenue and $4.4 billion in operating profit in 2025. Falcon9 had ~$4 billion in revenue, so they didn't cheat by subsidizing starlink with Falcon9.

Does Starlink pay SpaceX for launches?

For SpaceX it’s critical to maintain a steady cadence. If they don’t, they lose institutional knowledge. If the cadence drops below a point, their effectiveness at reusing rockets will also drop, defects will creep in, and they’ll have higher fleet churn.

Yes. Amount is actually irrelevant, since if they underprice it SpaceX loses launch revenue, and if they overprice it Starlink looks less profitable.

The more important takeaways are that SpaceX's near-monopoly launch business is profitable but not nearly as big as Starlink, and Starlink is a good business but not one to justify a trillion dollar valuation

Yeah, Starlink is about 3.5x the space launch revenue but still only about 0.5x in terms of profit. Falcon 9 is as optimized as a rocket could be, and absolutely owns the market. Starlink is a mostly rural service with global consumer pricing where average monthly rates in poorer countries drag the average down. Starlink government and commercial business, however, is growing quickly and I expect that soon Starlink will be ahead of launch, in terms of income, probably by the end of this year.