Hi Sarah –
1. Check out
Andrew's answer in this post. In my experience, people tend to use them interchangeably, even though they are slightly different. In many cases, Interval is perhaps the more conservative choice, which is reflected in its larger error ranges.
2. Date() is an incorrect way to summarize a phase. As Christopher
puts it here (see also the rest of the thread), Date() "is certainly no guide to the length of the phase". Instead, it's an "estimate for the date of an undated [event] within that phase." I usually use Date() when I have other priors, not just a phase. For example, if I have a depth sequence, I place Date() at a specific depth and get an estimate for an undated sample at that depth.
3. As for phase beginning and end, best practice is what you're doing, referring to starting and ending boundaries. Others use the queries First and Last (the first and last events in a phase – see the manual), which are often similar to, but not the same, as phase boundaries.
The best way to summarize a phase, as far as I can tell, is in Bronk Ramsey (2017:Fig 3) – on the same line, the two boundaries (green and red), the KDE in the middle, and the medians below. You can visualize this with View > Plot Stacks.
Hope this helps, Erik