1. You need to buy rechargable batteries. See what I do and why.
it's the best way to ensure no one is too lazy to keep the batteries charged. and keeps the protective silicone jacket on the remote.
2. NO. The wired one is brighter IR for the remote and costs money in batteries. If you plan to travel often with your wii then it could be handy. I own a wireless sensor bar ($6.00US after sipping from ebay) because I travel often and thought the sensor bar wire looked fragile.
3. You need to buy the component cables separately. I got mine from ebay for $6.81US after shipping... I've seen posts where someone found a source for $5.60US after shipping.
4. I got my 3rd party power brick from ebay for $18.00US after shipping.
my wireless sensor bar and brick are in a travel case. when i need to hit the road i don't pack any wires, just console, remotes, and games.