Is there a parking lot/structure under the museum?

No there is not one.but this one on the side by St.Regis but not connected. If you look on website it will show all garages close to the museum.

There is a large public parking structure 5th and Mission, across from the Metreon that works well. 2 blocks away.

Yes, 25$.

I personally rely on public transportation around there. A lot less hassle.

Thursday after 6pm(museum extends hours), or Sunday, You can park anywhere near museum on the side of big street with meters or small streets vertical to big streets, where there are no meters, for FREE! Short walks to the museum :) be aware of street cleaning you can’t park there.

Through the end of Feb 2018, you can get 25% off the first 3 hours at the MOMA garage -- see the front desk about it.

No but there is one next door