The answer to your Question 1 is NO. See Th. 5.1 on p. 2089 of my 2015 JPAA paper of (volume 219)my 2015 JPAA paper.
The answer to your Question 2 is YES. See 1963 Annals paper of Shimura ``On analytic families . . ."1963 Annals paper of Shimura "On analytic families ..".