User need to reload after login to be logged

I have a very odd problem. After user or admin is logged he need to reload page to be logged. Or to go to page which hasn’t been visited. I assume it is caching problem.
I’m using W3 Total Cache with my Customizr theme.
This problem is driving me crazy, I’m trying to fix it for days now, but nothing helped.
I tried different setups in W3TC, and nothing.
I posted a topic there, but got no answer, and sorry if this topic doesn’t belong here, but I’m really desperate now.

Please, help me. Support here was awesome (thank you very much d4z_c0nf for previous help).

P.S. I also have a problem with dropdown menu. It won’t show sometimes, and I assume it is connected with caching also.

Thanks again!