1) When doing a mash out you generally target around 168F, this is a temp that is quite a bit higher than mash temp, but still cool enough to be 'safe' without worrying about tannin extraction. You can do it different ways, but a lot of people prefer to use a particular volume of boiling water because it's easy to hit the temp exactly, and then all you need to calculate is the volume. Brewing software will do that for you, if you don't use software then I can't make any specific recommendation for how to calculate it, but I'm sure there's info out there.
2) The mash out is done after the 60 minute mash (or however long you mash) because it will halt conversion, which would be bad if it hasn't completed.
3) The mash-out water is no different than any other water you add, so yes, you do need to take it into account. I don't see why it would be a 'bad thing'...
4) The point of a mash out is to maximize solubility of the sugars. The more sugars you can dissolve into the wort, the more you can get out. Yes, this leaves less sugar in the mash for the sparge to remove, but evidence seems to suggest that overall, it improves efficiency. Since mash out stops conversion, it can "pin" the current state of mash conversion where it is so it doesn't change on you during the runoff, but I don't think this is an important reason for many of us.
I don't really know of any serious 'cons'. It's a little added effort, and requires a little more mash tun space, and I suppose if you don't stir it quickly after adding the very hot water you could get a 'hot spot' and have tannin extraction... But if you've got the space, do it properly, and don't mind the couple minutes of effort, I don't see much downside.