Sorting with select tag and checkbox staying checked

I have two questions:

1) I would like to sort an array of products with select tag using controller. This is the controller:

public async Task<ViewResult> Index(string sortOrder,List<int> checkedTypes)
    {
        var products = from p in _db.Products
        select p;
        ViewData["PriceDesc"] = string.IsNullOrEmpty(sortOrder) ? "priceDesc" : "";
        ViewData["NameDesc"] = string.IsNullOrEmpty(sortOrder) ? "nameDesc" : "";

        switch(sortOrder)
        {
            case "priceDesc":
            products = products.OrderByDescending(p => p.Price);
            break;
            case "nameDesc":
            products = products.OrderByDescending(p => p.Name);
            break;
            default:
            products = products.OrderBy(p => p.Name);
            break;
        }

        return View(products);
    }

When I used the code below, it works as intended:

<a asp-action="Index" asp-route-sortOrder="@ViewData["PriceDesc"]">
        <button >Price desc</button>
    </a>
    <a asp-action="Index" asp-route-sortOrder="@ViewData["NameDesc"]">
        <button >Name desc</button>
    </a>

However, I want the sorting to be displayed as a dropdown(with tag),and without submit button, something like this:

<select name="sorting" id="sorting">
            <option disabled selected>Select option</option>
            <a asp-action="Index" name="priceDesc" asp-route-sortOrder="@ViewData["PriceDesc"]">
                <option >Price: high to low</option>
            </a>
    <a asp-action="Index" name="nameDesc" asp-route-sortOrder="@ViewData["NameDesc"]">
                <option>Name: descending</option>
            </a>
        </select>

But for some reason the code above just ignore tag and the tag just disappear. And nothing happens when I choose one of the options.

2) I also have checkbox that works. But when the products are filtered with the checkboxes, the checkboxes stay unchecked. This is the controller code:

if(checkedTypes.Count != 0){
            products = products.Where(p => checkedTypes.Contains(p.TypeId));  
          
        }

and this is the view code:

 @foreach(var type in types)
            {  
                <input type="checkbox" name="checkedTypes" value="@type.id"/>
                <label>@type.Name</label>
                <br>    
            }   

Is there a way to keep the checked checkboxes stay checked even after the filtering on controller is done?