Homogeneize Date usages

The first commit replaces the use of timestamps in search with Date instances, to reduce the number of different formats we use to refer to dates (dates, date strings, moment instances, timestamps).

The second removes some unnecessary moment calls when I believe the native call is as short and we don't have to worry about how moment would do it. Also moment is quite heavy but we could consider using alternatives https://github.com/iamkun/dayjs for some remaining uses (adding/substracting days quickly and format dates mainly).