I do get about three bars of wifi with my laptop sitting like a weirdo in the corner of my sleepout which is indeed next to a big sliding door window so if i got this new wireless router it would make sense to put it there and then I would have full bars all around my room?
Yes, your laptop will display the signal strength as full bars since your second router is in the same room as you. Do note that the 2nd router will only get the same "3 bar" signal strength from your 1st router, so performance won't increase.

In fact, with wireless repeating, your performance will decrease by half. But really, connectivity is more of an issue than performance anyway.

This is what I get when I go onto my current wireless router/modem seen as its a hybrid checking to see if it supports repeating think so?
Set the 1st router in your parents' house to Access Point AND repeater mode. Set the 2nd router in your sleepout to Repeater Only.

I have no idea whether that router you showed in the second pic will work or not. Honestly, no idea. But since you said that your dynalink router can't really be bought anymore, you'll have to take a shot at something anyway.

edit: This product says that it will work with routers that don't have WDS functions. That suggests their normal models' repeater functions will work with other brands as long as they have WDS. (Which you do). Seems like a pretty safe bet for you linked router then. Or pay $11 more for surefire success. Your choice.