I suppose some of it would be lost to the vacuum of space, though it would have to be an impressively large space transport market for that to start making a global difference.
Also weird to think about the long-term implications of Earth's resources leaving Earth permanently/irrecoverably.
Carbon neutral assuming you are creating the CH4 using atmospheric CO2. On Earth I suspect that most CH4 comes from natural gas. On Mars it would be carbon neutral - but that's actually a bad thing, assuming that your goal is to increase the atmospheric density on Mars.
Yes, that was the assumption of the parent question. I expect that initially all the fuel for Earth launches will be sourced from natural gas, but if they start doing regular sub-orbital passenger flights they'd have to convert to renewable energy + atmospheric CO2, otherwise they'd just (partially) undo the work Tesla is doing.
I thought CH4 was ~four times more potent in the greenhouse effect than CO2, significantly worsening the global temperature rise.
The rocket works by burning the methane, though, instead of releasing it into the atmosphere. Burning it results in water and carbon dioxide.