You set DockPanel.Dock="Top" to the StackPanel, but the StackPanel is not a child of the DockPanel... the Border is. Your docking property is being ignored.
If you move DockPanel.Dock="Top" to the Border instead, both of your problems will be fixed :)
May be it will helpful:
<Border BorderBrush="Black" BorderThickness="1" HorizontalAlignment="Left" Height="160" Margin="10,55,0,0" VerticalAlignment="Top" Width="492"/>
What about this one :
<DockPanel Margin="8">
<Border CornerRadius="6" BorderBrush="Gray" Background="LightGray" BorderThickness="2" DockPanel.Dock="Top">
<StackPanel Orientation="Horizontal">
<TextBlock FontSize="14" Padding="0 0 8 0" HorizontalAlignment="Center" VerticalAlignment="Center">Search:</TextBlock>
<TextBox x:Name="txtSearchTerm" HorizontalAlignment="Center" VerticalAlignment="Center" />
<Image Source="lock.png" Width="32" Height="32" HorizontalAlignment="Center" VerticalAlignment="Center" />
</StackPanel>
</Border>
<StackPanel Orientation="Horizontal" DockPanel.Dock="Bottom" Height="25" />
</DockPanel>