[SalesForce] Can we modify/remove global header section from Lightning Pages

You can use facets to do this. For example, a simple test.app:

<aura:application access="GLOBAL">
    <aura:attribute name="messages" type="Aura.Component[]" access="GLOBAL"/>
    <div aura:id="messages">
        {!v.messages}
    </div>
    <ui:button label="Show Message" press="{!c.showMessage}"/>
</aura:application>

The corresponding testController.js:

({
    showMessage: function(component, event, helper) {
        $A.createComponent("ui:outputText",
                           {value: "Message created at " + Date.now()},
                           function(cmp) {
                               component.set("v.messages", [cmp]);
                               setTimeout(function() {
                                   $A.run(function() {
                                       component.set("v.messages", []);                                       
                                   });
                               }, 5000)
                           });
    }    
})

The facet is an attribute of type Aura.Component[] that you render in the desired spot using an expression such as {!v.messages}. When the element is created, set the value of this facet to be an array with one component, the [cmp] in the code. Set a timeout, etc., then set the value of this facet to be an empty array with []. Make sure to wrap the calls with $A.run.

If you use the Chrome or other tools to inspect the DOM, you'll see the component elements appear and disappear.

If this isn't what you're looking for, please provide more details.

I don't think there is an SLDS class specifically that enables scrolling, however an easy way to enable this using standard CSS is to do the following:

<style>
    .scrollable {
        height: 400px; <!-- Don't need this if it's okay for height to be dynamic -->
        overflow-y:auto;
    }
</style>

<div class="slds-page-header">
    <p class="slds-text-heading--label">Contacts</p>
    <h1>My Contacts</h1>
</div>
<div class="scrollable">
    <!-- The rest of the page goes here -->
</div>

EDIT: As @techbusinessman wrote in a comment below, there is now a standard utility class for making a div scrollable. The documentation can be found here: https://www.lightningdesignsystem.com/components/utilities/scrollable/#overview

The code would now be as follows:

<div class="slds-page-header">
    <p class="slds-text-heading--label">Contacts</p>
    <h1>My Contacts</h1>
</div>
<div class="slds-scrollable">
    <!-- The rest of the page goes here -->
</div>