If you only want to use the datepicker plugin of jQuery UI with EasyUI, yes, they can work together. You can find the example here:
http://jsfiddle.net/qX26t/

Some conflicts may occur when using easyui with jQuery UI together. Please try to use the datebox plugin of easyui.

When calling 'load' method to load form combobox, the 'setValue' method of combobox will be called to set the combobox with new value, the related item of drop-down will be selected too.
Please see this example http://jsfiddle.net/YBhB9/.

Make sure that you are using the newest datagrid-filter.js file. If not please update this newest file to your page. After that, if your issue continues, please provide an example to demonstrate your issue.

After update a row, you need to call .linkbutton() to create the linkbutton again.

Code:

$('#dg').datagrid('getPanel').find('a.easyui-linkbutton').linkbutton();

The 'formatter' function only returns the formatted string. This function can't create any components automatically. You have to construct them manually. Please try to call .linkbutton(...) when loaded data successfully.

Code:

$('#dg').datagrid({
  onLoadSuccess:function(){
    $(this).datagrid('getPanel').find('a.easyui-linkbutton').linkbutton();
  }
});

The local searching is similar to this tutorial. All you need to do is to filter the 'data' and call 'loadData' method to render the new rows. Please see the 'doSearch' function to learn how to achieve this functionality.

Code:

function doSearch(q){
    var rows = [];
    $.map(data, function(row){
        for(var p in row){
            var v = row[p];
            if (String(v).indexOf(q) >= 0){
                rows.push(row);
                break;
            }
        }
    });
    $('#dg').datagrid('loadData', rows);
}

Here is the tutorial shows how to add search functionality.
http://www.jeasyui.com/tutorial/datagrid/datagrid24.php

The appendRow method append a new row to the current page. To append to other page, you have to change the grid page before calling this method.

You have to use draggable and droppable plugins to achieve this functionality manually.

To update and apply column properties, you have to recreate datagrid with the new columns.

Code:

var dg = $('#dg');
var columns = dg.datagrid('options').columns;
columns[0][2].title = 'new title';  //change columns ...
dg.datagrid({columns:columns});  // recreate datagrid

Please confirm if you are using the latest edatagrid plugin. If not, please download it from http://www.jeasyui.com/extension/edatagrid.php.

Please download the latest 'datagrid-filter.js' file from http://www.jeasyui.com/extension/datagrid_filter.php. You may need to override the filter operators. The operator code looks like this:

Code:

$.extend($.fn.datagrid.defaults.operators,{
less: {
text: 'Less',
isMatch: function(source, value){
return source < value;
}
}
})

All the $.messager functions work in async mode. This means that the user can't block it before doing any more. To make the confirm message window works like a native confirm window in 'onBeforeClose' event, please try the code below:

Code:

$('#win').window({
  onBeforeClose:function(){
    var p = $(this);
    $.messager.confirm('Confirm','Are you sure you want to close?',function(r){
      if (r){
        var opts = p.panel('options');
        var onBeforeClose = opts.onBeforeClose;
        opts.onBeforeClose = function(){};
        p.panel('close');
        opts.onBeforeClose = onBeforeClose;
      }
    });
    return false;
  }
})

If a data item has 'selected' property set to true, it will be selected when loaded successfully. All you need to do is to set 'selected' property for the specified item from your returned json data. Look at the following data, the 'Ruby' item will be selected since its 'selected' property is set to true.

Code:

[{
"value":1,
"text":"Java"
},{
"value":2,
"text":"C#"
},{
"value":3,
"text":"Ruby",
"selected":true
}]