Transfering WP from one hosting company to my own dedicated server

Hello everyone.

I need help. I have problem with my WP site after transfering it to another server. I used cPanel file manager, tar all files in /home/user-dir/public_html/wp-site (shared hosting) and downloaded it to my desktop. After that I did backup of my database with phpMyAdmin and downloaded it to my desktop too. After downloading sql file, I zipped it into one smaller file with extension xxx.sql.zip (it was bigger then 2Mb).

I transfered tar file with all of my files in it to a new server on location /var/www/html/user-dir via ftp. Used ssh access on new dedicated server and extracted it. Created database and database user with same name and password like on old hosting. Imported .sql.zip into database without any error. Changed DNS settings and after 15 minutes, my WP sites go live.. but I could open only home page ??
Any other links to my pages and posts does not work. I think it is problem with permalinks and because of that, I repeated all the process above for one more time but before backing up – I set Permalink to default and just in case – deactivated all of my plugins.
After transfer and all of that job again – all links are working, but when I tried to activate Permalinks again (setting them to: https://wp-site.com//%postname%/), they give me error 404 like before ?? I need that permalinks because other sites are linked to my links.

I opened /var/log/httpd/ file and getting this error (some user details changed for privacy):

[error] [client xx.xx.xx.xx] File does not exist: /var/www/html/user/category, referer: https://some-site.com/
[error] [client xx.xx.xx.xx] File does not exist: /var/www/html/user/page, referer: https://some-site.com/

Another thing make me confused. I have dir “icons” in my site root dir with some .png pictures in it. Now when I try to access it via: https://some-site.com/icons/picture1.png I get this error:
[error] [client xx.xx.xx.xx] File does not exist: /var/www/icons/picture1.png

I have mod_rewrite active in httpd and my virtual host settings are same like on other working sites (documentroot are pointing to /var/www/html/user). Using Centos 6.5 x64. Tried all possible changes in .htaccess but with no results.

I am really confused and did not get this kind of headache for a long time and I transfered lot of sites in past few years without any problems with WP sites.
If somebody have some idea where I`m wrong – I would appreciate your help.

Thanks in front!