You can definitely do this within Jira Align.
As far as rights, the System Role in Jira Align has to allow someone to change the Initiative. (If you can't save the initiative record, you won't be able to add tickets to it.)
As far as multiple projects, it is important to remember that an Align program or team of teams can draw from many Jira projects. This is all explained in the Best Practices session here: https://community.atlassian.com/t5/Jira-Align-articles/Jira-Align-and-Jira-Integration-Best-Practices-and-FAQ-Table-of/ba-p/1253497
In Jira Align, an Initiative can be worked on by 1 to many programs. So Jira tickets from any of the programs that contribute to the Initiative can be connected to the Initiative.
Important caveat #1: This can be a bit confusing because different enterprises use different terminology. In the case above, let's assume you have:
Initiatives are composed of one to many Jira Epics
Jira Epics are composed of one to many Stories
Stories have 0 to many sub-tasks
In Jira Align, you can also introduce a Capability layer between the Initiative and the Epic. This Capability object is rarely used, so we'll assume you have just Initiative, Epic, Story, Sub-task.
Important caveat #2: Although a program in Jira Align can be connected to many Jira projects, all the projects for a given program must be in the same Jira Instance.
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