@AidezMoi:
1) center my logo
— Add the following css in “Appearance => Theme Options => Custom CSS” box.
#header-left, #site-logo { width: 100%; }
#site-logo a { text-align: center; }
#hgroup-wrap #site-logo img { display: inline; }
2) put the search bar and social icons on the same line
— Add the following css in “Appearance => Theme Options => Custom CSS” box.
#header-right { max-width: 100%; width: 100%; }
#header-right .widget { clear: none; margin-left: 15px; }
3) put the social icons and search bar above the logo on the right
— To make this you need to upgrade to pro version, where there is header top sidebar. There you can add in Social Widgets and Search Widget.