1. Do I have to restart JIRA when I change a template?
No, no es necesario.
2. Can you modify JIRA default templates?
No, solo se pueden modificar las plantillas que hayas generado con este add-on.
3. Do templates store in JIRA database? I mean, If I upgrade JIRA do I lose the templates or I keep them?
Yes, they are stored in JIRA database and you will keep then if JIRA or the addon are upgraded.