1 ) When you 'minimize' a page you cannot see it. You probably mean 'restore down'. In any case, you seem to be talking about what happens when a page is viewed through different sized windows. Similar things happen when a page is viewed at different resolutions and/or text sizes. The best solution is a more flexible design that allows your content to flow into the space alloted for it with any overflow being able to be scrolled into view vertically. There are other approaches.
2 ) I'm not aware of any particular element which is called a 'search box'. However, if you want to place an image in the middle of something, you can do it at least two ways:
HTML Code:
<div style="text-align:center;"><img src="whatever.jpg"></div>
or:
HTML Code:
<div><img style="width:50px;height:20px;margin:0 auto;" src="whatever.jpg"></div>
These both assume that the division is of the dimensions you mentioned.
Note: Styles should be in a stylesheet.
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