From the looks of it you are simply trying to get the labeledit control to work and you are not making your own custom callbacks. correct?
Assuming the above statement is true I will attempt to answer your questions.
1. No, the control exposes an event (UpdateLabel) that you simply need to write a handler for.
2. I have not tried the control in a repeater, but don't anticipate it to work yet. I say this because the event lifecycle currently stops for a callback at the page init. I am assuming your grid is not bound until later. Note: The callbackFail method does not contain any indication of an error. It should have an alert in it to notify you that your control was not found. To see this open up the dnn.controls.dnnlabeledit.js file and update the callbackFail function to this.
callBackFail: function (result, ctx)
{
alert(result);
ctx.cancelEdit();
},
I say that it will not work yet, because I am in the process of enhancing the Callback capabilities to allow it to continue through the page's lifecycle. I blogged about how this is being incorperated into the new DNNTabStrip control here. I see no reason as to why this could not be added to the labeledit control.
To answer your question on figuring out the row index, the quick answer is no. However, depending on how you register your event (see 3) or how you can interpret the ID assigned to the control it may be possible.
3. This of course is tricky part, for what needs to happen is you need to associate an event handler to each instance of the control.
4. This is something I am currently working on. There will be two ways to handle this. The first is to offer a BeforeSaveFunction that you can write custom code in that will return true or false determining whether or not you wish to cancel the update. I envision a dialog box here. The second is to offer a toolbar. I have a prototype shown here.
5. Should be possible with CSS
To summarize, being able to use the control inside a repeater is really not possible without some major hacks right now. However, in the near future it should be possible. I most likely will create a sample to demonstrate this capability.