Hey please check your logo >> http://take.ms/V6uop you are using a transparent logo which is fine but you are ignoring extra space on right side. I have placed a black background in your logo to show you extra space your logo have on right side. Please remove that space using photoshop and then re upload your logo without any space in left, right, top or bottom.
It will increase the size a little bigger.
Second thing you can do to increase the width of logo please go to >> wp-admin >> appearance >> customizer >> additional CSS and add the following CSS changing the height you can adjust the size of logo. Please note you can go to a specefic height only otherwise logo will start stretching reason is available space of width, logo height increasing will adjust width simaltaniously as well. So keep changing number of height and see what looks perfect.
.header .logo img {
width:auto;
height:45px !important;
}
In above code you can start changing 45 to your numbers theme's default height is 40px thanks.
To remove search icon please go to wp-admin >> appearance >> customize >> additional CSS and add following CSS make sure you add this CSS at end of any existing CSS there after any closing } if there is no CSS available already you can write it on top.
.header .search-wrap.float-right {
display:none !important;
}