Set the value property of the SelectItem to the ID of your entity not its instance.

List<SelectItem> listForYourCombo = new ArrayList<SelectItem>
for(YourEntity entity : list) {
SelectItem si = new SelectItem();
si.setLabel(entity.toString());
si.setValue(entity.getId()); // <==== The point is here DO NOT USE si.setValue(entity)
}

The you can bind the value property of selectOneMenu to an Integer represented the selected  item using the ID of your entity in your back bean.

<f:selectOneMenu value="#{yourbackbean.selectedEntityID}" />

if you do not underestand the case, download NetBeans IDE you can find wonderful samples there to work with JSF

At first you should have an action (Managed Bean) object with one of these scopes :
1- conversation
2- session
3- application

and then you could use ajax base thechnology for better performance and efficiency for Rerendering page  without refresh it like Richfaces.

So then you should bind a field or variable from that action (Managed Bean) with EL to JSF page at combo box that you use like this :

<h:selectOneMenu id="dropdown-1" value="#{action.field}">
<f:selectItems value="#{helper.items}"/>
</h:selectOneMenu>

and in this case value of selected  will survive till action is live at those scope.

I think the problem is that you set your Backing Bean Scope to Request. Try to
change it to Session and you'll see it saves the states of your DropDownLists
after submitting the page. Here is a very simple code  I use in my projects. As I
mentioned the BackingBean has Sessio scope:

<h:selectOneMenu binding="#{editExpertAddressInfoBackingBean.province}"
style="width: 160px"
valueChangeListener="#{editExpertAddressInfoBackingBean.provinceChanged}"
onchange="submit();">
<f:selectItem itemValue="" itemLabel="" />
<f:selectItems value="#{editExpertAddressInfoBackingBean.provinceItems}"/>
</h:selectOneMenu>


<h:selectOneMenu binding="#{editExpertAddressInfoBackingBean.city}"
style="width: 160px">
<f:selectItem itemValue="" itemLabel=""/>
<f:selectItems value="#{editExpertAddressInfoBackingBean.cityItems}"/>
</h:selectOneMenu>





public void provinceChanged(ValueChangeEvent e) throws AbortProcessingException
{
if (province.getValue() != null &&
!"".equals(String.valueOf(province.getValue()))){
Long id = Long.valueOf(String.valueOf(province.getValue()));
cityItems = loadCities(id);
} else{
cityItems.clear();
}
}

No the problem is that you must
override toString() method of your entity because the selected  
value key is the string value of your object.

In HTML elements the java objects are not known, and they are string, array of string.

For more information refer to JSF specs, and for beginners samples in NetBeans IDE.

I think that it's so usual to lost values  when you submit your form and change request .
Maybe it's better , use of Ajax submit, instead of submitting form when you change the value of dropdown  . it cause to send a request to server and change a part of page  which you want but not your whole page . for doing this that's enough to use form tag of Ajax for JSF lib(a4j) and add
" ajaxSubmit="true" "
to your form tag in the page.
for more information refer to A4J lib refrence.