Align div inside parent div & padding not working

Your menu structure should be made up of a unordered list, like so:

<div class="menu">
    <ul>
        <li class="Div1"><a href="#chi-siamo">Chi Siamo</a></li>
        <li class="Div2"><a href="#contatti">Dove Trovarci</a></li>
        <li class="Div3"><a href="#servizi">Servizi Investigativi</a></li>
        <li class="Div4"><a href="#formazione">Formazione</a></li>
        <li class="Div5"><a href="#news">News</a></li>
    </ul>
</div>

Which you can simply style with:

.menu {
    width:100%;
    text-align: center;
    box-shadow: 0px 0px 6px 2px #D8D9DA;
}

.menu ul {
    border: 1px solid #000;
}

.menu li { 
    display: inline-block; 
    padding: 10px 15px;
}

You can control responsive styles like this:

@media screen and (max-width: 620px) {
    .menu li { 
        padding: 25px; 
    }  
}

I have removed the padding and border away from the menu class as you want this to be 100% wide, the border has been applied to the ul with the padding now being applied to each li instead.

JSFiddle

Hope this helps.