Can I have two date pickers in one field (i.e. a proposed date and then an actual date)

[TNick]

If not, how do I get the two separate fields to appear side by side in an issue?

[Nic Brough -Adaptavist-]

No, date fields all hold one single item, you'll need two of them. 

As for the layout, again, no, JIRA lists them one line at a time.

Both of those could be done if you're willing to find or write an add-on to provide a dual field, and/or code to re-arrange the display