Based on above, looks you are struggling to have a handle on dropdown inside Grid via jQuery/JavaScript.
For ASP.NET, you would have something like:
<asp:TemplateField HeaderText="Country" ItemStyle-Width="150">
<ItemTemplate>
<%# Eval("Country") %>
</ItemTemplate>
<EditItemTemplate>
<asp:DropDownList ID="ddlCountries" runat="server">
<asp:ListItem Text="Please select" Value="" />
<asp:ListItem Text="United States" Value="United States" />
<asp:ListItem Text="India" Value="India" />
<asp:ListItem Text="France" Value="France" />
<asp:ListItem Text="Russia" Value="Russia" />
</asp:DropDownList>
</EditItemTemplate>
</asp:TemplateField>
Following references will help you move:
Find and validate TextBox and DropDownList controls in GridView row using jQuery in ASP.Net[
^]
How to get Dropdown selected value inside GridView using JavascriptEverything Technical[
^]
JQuery: Fill DropDown and Show Records in GridView Format in ASP.Net[
^]
Try out, they are with complete examples.