I have those problems with Nectar Slider, I tested and it happens with your salient-blog demo too:

1. If I click the right sidebar button, close it, click search button, the dropdown menu (which is for small screen) shows up instead.
2. On retina screen, when I scroll down, the logo on menu bar disappears. It does not happen on normal screen. I checked and see you didn't add the retina-logo to original span in nectarslider.js, plus you didn't hide the the non-retina logo for small screen in responsive.js, so I have to edit them both.
3. On retina screen, if I open the home page (with nectar slider) in small mode (menu with centered logo) then expand the window to switch to transparent menu, the logo disappears. If I open in full mode initially, the logo shows up. It does not happen with other page so it maybe nectarslider bug. I checked and found out that a#logo width is not set correctly on retina screen. On retina screen, it is always 0.

I'm using the latest version from themeforest (7.5.0.2). Unfortunately haven't published the website yet.

However I'm sure that I'm using latest files 7.5.0.2 and the bugs I reported are the same as your salient-blog. I checked for updates from themeforest but no update is found. I'm trying to solve those issues on my localhost.

Here is my progress:

#2: Solved

In nectarslider.js, function directionalNavColorInit(), you only add img:first to .original, but doesn't add .retina-logo -> Add retina-logo to .original in directionalNavColorInit().

Then, in responsive.css:

@media only screen and (-webkit-min-device-pixel-ratio: 1.5) and (max-width : 1000px), only screen and (min-device-pixel-ratio: 1.5) and (max-width : 1000px), only screen and (min-resolution: 144dpi) and (max-width : 1000px) {
    #header-outer.transparent.directional-nav-effect header#top #logo span.original img.default-logo {
        display: block!important;
    }
}

Set 

display:none#important 

to hide the non-retina default-logo on retina screens.

#3: Solved

- logo doesn't show when switching from mobile to non-mobile because #logo is set to 0 because of this nectar-slider.js. For some reason $('.retina-logo').width() returns 0.

//retina logo fix
if($('.retina-logo:visible').length > 0) $($selector+' header#top #logo').css('width',$('.retina-logo').width());

Remove the fix solves the problem.

- Switching from mobile to non-mobile doesn't update the header skin correctly based on current slide because you only update the width on non-mobile mode (nectar-slider.js):

                    if($('body.mobile').length == 0) { 
                        $('#header-outer nav > ul.sf-menu > li > a, #header-outer nav > ul.buttons > li#search-btn a, #header-outer nav > ul.buttons > li.slide-out-widget-area-toggle a > span, #header-outer .cart-icon-wrap, header#top #logo').each(function(){
                            var $itemWidth = parseInt($(this).find('span.original').attr('data-w'));
                            $(this).find('span.light').css('display','inline').transition({'margin-left': '0'},0).css('display','inline').transition({'width' :  $itemWidth }, 325, 'easeInOutQuart');
                            $(this).find('span.light > span').transition({'margin-left': '0'},0);
                            $(this).find('span.dark').css('display','inline-block').transition({'margin-left': $itemWidth, 'width' :  '0' }, 325, 'easeInOutQuart');
                            $(this).find('span.dark > span').transition({'margin-left':  '-' + ($itemWidth) + 'px', 'width' :  $itemWidth }, 325, 'easeInOutQuart');
                        });
                    }

To prepare for switching to non-mobile mode, update the width too:

                    else{
                        $('#header-outer nav > ul.sf-menu > li > a, #header-outer nav > ul.buttons > li#search-btn a, #header-outer nav > ul.buttons > li.slide-out-widget-area-toggle a > span, #header-outer .cart-icon-wrap, header#top #logo').each(function(){
                            $(this).find('span.dark').css({'display':'inline','width' :  '0'});
                            $(this).find('span.light').css({'display':'inline-block', 'width' :  "" });
                        });
                    }

#3: Solved

In Init.js, when user clicked on normal sidebar button, you mistakenly change class of the hidden mobile menu

    $('.slide-out-widget-area-toggle > div > a').removeClass('closed').addClass('open');
    $('.slide-out-widget-area-toggle > div > a').addClass('animating');

Adding :not(.std-menu) solves the problem

    $('.slide-out-widget-area-toggle:not(.std-menu) > div > a').removeClass('closed').addClass('open');
    $('.slide-out-widget-area-toggle:not(.std-menu) > div > a').addClass('animating');

After the fix, the #toggle-nav only shows "closed" but it works as intended.