Center menu and logo

It looks like you’ve already managed to get your navigation centered, but try this for logo:

.header-wrapper {
    display: block;
    float: none;
    width: 100%;
}
.header-wrapper a[rel^=home] {
    display: block;
    max-width: 150px; /* your logo's desired max-width here*/
    margin: 0 auto;
}

and for navigation, which you already seem to have solved:

.main-navigation {
    float: none;
    width: 100%;
}
.main-navigation .menu > ul {
    text-align: center;
}

The hover seems to be applied to the whole header area with this block of code from original theme:

.site-header {
	clear: both;
	display: block;
	min-height: 30px;
	padding: 30px 0;
	position: relative;
	transition: opacity .5s ease-in-out;
	-webkit-transition: opacity .5s ease-in-out;
	-moz-transition: opacity .5s ease-in-out;
	-o-transition: opacity .5s ease-in-out;
	width: 100%;
}
.site-header:hover,
.site-header:focus {
	opacity: 1;
	transition: opacity .5s ease-in-out;
	-webkit-transition: opacity .5s ease-in-out;
	-moz-transition: opacity .5s ease-in-out;
	-o-transition: opacity .5s ease-in-out;
}

If we wanted to remove the entire effect from the header this would work:

.site-header {
	opacity: 1;
	transition: none;
	-webkit-transition: none;
	-moz-transition: none;
	-o-transition: none;
}

However, it sounds like you just want to remove the hover effect from the navigation, but leave on the logo. So we would have to be a little more elaborate in customizing, try:

.site-header {
	opacity: 1;
	transition: none;
	-webkit-transition: none;
	-moz-transition: none;
	-o-transition: none;
}
.header-wrapper {
        opacity: 0.5;
	transition: opacity .5s ease-in-out;
	-webkit-transition: opacity .5s ease-in-out;
	-moz-transition: opacity .5s ease-in-out;
	-o-transition: opacity .5s ease-in-out;
}
.header-wrapper:hover,
.header-wrapper:focus {
	opacity: 1;
	transition: opacity .5s ease-in-out;
	-webkit-transition: opacity .5s ease-in-out;
	-moz-transition: opacity .5s ease-in-out;
	-o-transition: opacity .5s ease-in-out;
}