I can't say I know what's best for mobile...but I'd also support any updates or features focused specifically on mobile view/ease of use/etc...
I don't shop on a phone, but apparently people browse a ton using one. Having the search bar defaulted would be easier for me since I didn't know you need to click on those lines to get it. It's hard to really "browse" on a small phone screen IMO, but that seems to be where everyone is headed...
Maybe specific views or front page just for mobile would be nice...no idea honestly.
Also, when I look at the demo Abantecart site with my phone (Note 3), the Abantecart logo is under the lines button when in the horizontal (Portrait) view. Is there a way to fix this? On the vertical view, it looks ok. Happens with chrome browser.