@daftvader - quite a shopping list. Do you want a laptop or a desktop machine, how do you watch (or want to) the tv type stuff ? Gaming is a bit open ended as some of the latest stuff needs pretty high tech hardware.
I am sure there will be a pc equivalent at lower price but as I'm a fully signed up Apple boy here is "their" solution
iPad (£450-ish) for most of basic stuff, YouTube etc lots of fun games
Apple TV (£100) for streaming stuff onto the tv either from web (inc YouTube, netflix etc) or from your computer
MacMini (£500 or less second hand) connected to your existing screen/keyboard/mouse as your desktop machine, can run more complex games but not the most graphic intensive
This solution means multi users so everyone is happy.
MacBook pro (£1200) and connect it to tv with hdmi out for watching stuff on big screen - you can definitely do this with a windows laptop for half this price too. Letting junior get hold of a MacBook might not be smart as he/she may abuse it.