Hi, I'm a project manager who uses JIRA extensively (and PMP, PMI-ACP, Professional Scrum Master). There is an add-on for JIRA called Portfolio to help with many of your requirements.
For question #1, your product manager and project teams would plan out their projects in JIRA by identifying epics and stories as they normally would. The teams should use 'story points' to assess relative complexity for each story, we normally use fibonacci numbers. Then the teams should determine sprint lengths and their sprint capacity (story points per sprint) for the initial sprints. Unfortunately issues appear the same size regardless of the number of story points assigned (but you can see the numbers assigned to each issue) - you can see a visual depiction of the volume of work in each sprint using Portfolio. Ensure the first sprint is 'started' in JIRA before trying out the portfolio view below. Create placeholder sprints in JIRA for all the backlog story items in all projects, dividing them up by sprint in logical order.
Subscribe to the Portfolio add-on. Create a portfolio view, add all your projects, then select the capacity filter from the drop-down on the right-hand side of your screen. It should show your projects as horizontal swimlanes, with a bar graph representing the volume of work in each sprint. The green represents work that falls within the team's estimated capacity and red represents work that exceeds the team's capacity. Initially all of your issues and epics will appear grey, you can fix that by adding themes and assigning colours to the themes but I'll save that for another question.
For question #2, you can see that status of a project by looking at the JIRA board, the portfolio roadmap, and the reports that can be generated by JIRA and Portfolio. It won't remind you if you haven't checked in less than two days ago, however, you could just schedule in your calendar to listen in to the daily scrum call every second day.
For question #3, there is a different filter in the Portfolio software called "Target Schedule". If you select that filter, it will show any deadlines entered into the epics and stories. Using JIRA and scrum planning should help avoid last minute urgent issues (i.e. your due tomorrow example), your product manager should know at the end of each sprint planning session what to expect by the end of each sprint.
I'd be happy to walk through a demo of this during a call.