Do I have to purchase a 3D receiver? Surround Sound; 5.1 or 7.1 or something else?
You don't have to purchase anything.
What you want to purchase and what you can afford and what level of effort you want to put in are all factors in determining what you will end up wanting to buy. This is a long process, requires a level of expectation, as well as a budget. Room dimensions, and desire and ability by you to do the A/V work which is required.
For example, if you own the home, then you may very well want to run all your wiring behind the walls, and put in some good in-walls for the surrounds, and put in some decent floor standing loudspeakers. If you are renting, or have neighbors nearby, you may need to keep the volume much lower. A good subwoofer is always a solid purchase, but a good sub is pricey. All of those things MUST be connected to a surround receiver and if you want 3D then it has to support 3D (typically).
You mention mid-priced... Keep in mind that many companies consider a mid-priced A/V receiver to be $800-$1,000. They start at $200-$500 and the expensive ones are $1,200 to far more. A mid-priced subwoofer is often $800-$1,200. So, if that is 'expensive' in your mind, then you may need to adjust your expectations downward and let us know as there are deals to be had on 'entry-level' setups as well. But we need to know.
I cannot connect my XBOX to the LG's AV/Component connection. I've never seen this type (see image). Where do I go from here? I'm still using a first edition white XBOX {{{old}}}
As mentioned above, there is an adapter. The connection on your TV is just a way for LG to save space and money when they sell you that TV. Samsung typically ships WITH a breakout cable when they do the same thing, but they've been known to not include the breakout cable as well. So, you buy the adapter, and you will have your standard red, green, and blue component video connections as well as a red and white audio connection.
Of course, if you buy a receiver which upconverts all analog signals to HDMI, then you won't need that cable at all.
I'd love to be able to connect my new Macbook to my TV wirelessly. Is that possible?
I'm going with no.
Computer video sent wirelessly anywhere without a good wireless HDMI device tends to look terrible, be very laggy, and not acceptable. One HDMI cable from your laptop to the display makes everything perfect and reliable, it is the only way to go in my experience. When I hear about a solution to this request that I've heard dozens of times, I will let people know about it, but much like all other HD video, computer video is extremely high bandwidth and can't be sent wirelessly reliably and cleanly.