I think I got it.
First of all I restarted PC, after that had to set up security once again, because it was broken. Then recreated web addaptor with admin access and user from LDAP (now it worked, but before restart it didn't). And then checked web adaptors WEB-INF/web.xml file and found out that there is a line
<security-constraint>
<web-resource-collection>
<web-resource-name></web-resource-name>
<url-pattern>/services/*</url-pattern>
had to change /services/* to /* and that's all. All that maybe is because ArcCatalog maybe is using REST directory to get info about cache?
Why that restart changed security settings? Who knows ...