So, the Skype problems. The first one, which I'm not sure Riot can fix, is that sometimes receiving calls will forcefully tab you out after half a minute or so.

The other one is, to me, much more annoying. When a Skype notification pops up ("Name has logged int", "Chat has a new message", it creates a certain space on the corner of your minimap. There is no graphical differentiation, but if you click on this space you are tabbed out to Skype, and you have to tab back in. I have learnt to avoid this in the middle of clicking the minimap, because it changes your cursor to your normal cursor when you hover over the sweet spot. I right click outside the space and drag to where I want to go, then quickly click the spot and tab back in.

I don't know if these are things Riot can fix, or if it's something you need to do with Skype.

thats just how skype works you either learn to deal with it or use something else like ventrillo

Turn off notifications to fix the minimap issue.

You can try setting yourself to the red " Do not disturb" while you play skype. That way you wont get notification that you're being called, or that someone logged in or out. Skype is a very needy program and demands attention unless you put it in time out. This has nothing to do with League, and the best bet is to just temporarily disable the ability to receive the notifications.