It is said in manual to CompactTags portlet, that user should "open preferences interface" after portlet installation (said here).
If I press wrench icon, I get only "Look and Feel" and "Configuration" entries.
Is this mean that something wrong with my theme?
in your portlet.xml, you'll have to support the portlet's "edit" mode. That's how the JSR-286 standard names the "Preferences" setting. IMHO Liferay's choice of "Preferences" is a better name than "edit", thus it's used on the user-facing interface.
In addition, the user in question must have permission to "edit" the portlet.
partial portlet.xml: