Reconstructing hidden geometric structures in the data

Two important problems frequently arising in applications of statistical data analysis can be informally stated as follows.
(1) Distances between data points taken from some (unknown) manifold are measured. Can one reconstruct this manifold or its embedding in a given (say, Euclidean) space knowing just the information on the distances?
Can one get some information on its topology (e.g. Betti numbers, Euler characteristic etc)?
(2) Location of data points on an unknown manifold embedded in a Euclidean space are known with some errors. Can one reconstruct their ``true'' locations and the manifold itself?
Both problems are quite common to data science/big data and constitute what is nowadays known under the name ``manifold learning''.
We will discuss these problems (with the emphasis on (1), maybe just sketching (2), their motivation from applications, as well as their ``close relatives'' from metric geometry, both in "continuous" and "discrete" versions.