> Am I reading that correctly? For $50 you get ~1,000 transistors worth of die space, and it comes in a package?

That's correct, although it's $100: $50 for the space on the group-run chip and $50 for the physical stuff: the chip and the dev kit.

> Are there any other limitations?

Yes. Although, fewer and fewer with each next iteration.

> Full-custom analog design if you wanted?

For analog design, you would need to deal with https://efabless.com/ directly, pay $10k and get 100 QFN chips back with your design some time later.

TinyTapeout is digital-only, for now.