Why would anybody want non-informative distributions?

To have a prior distribution to use when very little is known about the estimand. It's meant to somehow capture the notion of minimal prior knowledge contributing to the posterior distribution, so that the data drive the conclusions, not the prior.

I don't know what it means for a confidence interval to be asymptotically valid.

The confidence coverage of a posterior interval is equal to the posterior probability mass of the interval plus a term which goes to zero as the amount of data increases without bound.

if your model of the data-generating process contains more than one scalar parameter...

E.g., a regression with more than one predictor. Each predictor has its own coefficient, so the model of the data-generating process contains more than one scalar parameter.