When I give myself OP via the console, she is given, but she is absent.
Without plugin of LuckPerms OP is given and functions normally.

Why do you even try to give yourself OP when plugin tells you that it will not work?
Do /lp user <username> permission set minecraft.command true if you want access to minecraft default commands..

I have a privilege on the server with OP. But it is not issued because of the LP. The players enter the /whois NICK command and see that they do not have an OP and ask me why it does not exist. There should be at least an OP visibility

If you're running a Sponge server (idk, I'm just guessing as you didn't give any information in the issue) - a players OP status has no effect on permission checks.

On Bukkit, OP status is considered, however it is considered AFTER permissions. You can use the autoop permission/feature if you want to control OP status with LuckPerms.

Aside from that, I don't really know what the problem is? What are you asking for exactly?