The answer to your Question 1 is NO. See Th. 5.1  of [my 2015 JPAA paper](http://arxiv.org/abs/1312.0377).

The answer to your Question 2 is YES. See [1963 Annals paper of Shimura "On analytic families .."]( http://www.jstor.org/stable/1970507?origin=crossref&seq=1#page_scan_tab_contents).