I can't login since I installed JFBConnect

Test: Facebook Javascript library inclusion
Fail: Another Facebook extension is installed, including the Facebook Javascript library, and may conflict.
I disabled all the plugins with facebook: System
- JO K2 Auto Post Facebook
- DP Calendar - Facebook

There is still something adding the deprecated Facebook all.js script on the site and it is conflicting with the new sdk.js that Facebook requires. You will need to continue looking to find what is still inserting this. If both all.js and sdk.js are on the site, Facebook authentication/likes/etc will not work.

When I go to clubinefbcn.cat/perfil-usuari I can't login with the joomla login. Also I can't login with the k2 login on the right.
- I saw the message with a Registered user. "You can not access the private section of this site"

I suspect that this is not related to JFBConnect. To test this, can you disable the 'Authentication - JFBConnect Social Integration', 'System - JFBCSystem', and 'User - JFBConnect Social Integration' plugins. This will render JFBConnect disabled. Then try to see if you have the same issues with login and registration. I couldn't register a new Joomla user either.

When I log with FB, I saw the message OK but it's not really OK because I saw a Register Form. When I click on the register form, it fails.
- With Google+ it seems similar.

It sounds like you have the 'Enable Automatic Registration' setting disabled. When disabled, JFBConnect will bring you to a mini-registration page. If you'd like JFBConnect to create the user automatically, without any prompting or ability to edit username, password, etc, then set the Automatic Registration on.

WIth JFBConnect » Open Graph Activity I saw "Invalid Token"

I cannot tell if you mean that this happens from the front-end or backend. If the front-end, then I would fully expect this, as you are having troubles authenticating and logging in.

I don't selected cache mode on general settings

In addition to the caching settings in JFBConnect > Configuration > General and the SCLogin module > Advanced section, you should verify that caching is disabled in the Joomla Global Configuration area and the System Cache Plugin is also disabled in order to test.

This should hopefully help to narrow down whether JFBConnect is contributing to the problem or not.

Thanks,
-Melissa