1. On September 10th, I'll be taking a flight from LAX to LHR. Would the pilot get on an initial heading of 341 after taking off, and then follow a great circle to LHR?
No, not unless he/she is planning on flying to Alaska! What you have done here is to enter the route in a "Great circle mapper" and then mis-read the track of 034.1 degrees as 341 degrees.
Since I know a good deal about flight instruments and the like, I'm worried that I might scare my CFI by keeping my head glued to the panel, never looking outside. Like, once my CFI believes I'm competent enough to pretty much fly the plane, but before s/he signs me off to fly solo, let's say we make a flight from LGB to LAX. After taking off, I'll tune into LAX's VOR on 111.10, set my HSI's heading bug to 101 (the radial connecting LGB and LAX) to remind me of the aircraft's course, and use my flight instruments to keep on the radial. Then I'll contact LAX's ATC when 5 nmi away, and request landing clearance, and follow their instructions to land.
Based on your first question, your knowledge of "flight instruments and the like" may not be quite as "good deal" as you first thought. You will (as others have said) be taught to fly visually. That means using your eyes and keeping a good lookout. In fact, in that part of the world an
excellent lookout. There have been some of the worlds worst mid air collisions in that small part of the planet, caused by pilots who failed to do just that (San Diego 1978, Los Cerritos 1986.)
If you take off from LGB - LAX with a HSI heading bug of 101 degrees, you will be flying almost in the opposite direction as the course inbound is 300 degrees.
3. Related to the above, let's say the aircraft has a TAS of 150 kts, and there's a 20 kt cross-wind at 045, causing our heading to be 095, causing the CDI to move 3 dots to the right. Would I apply right rudder to turn six degrees, and get the CDI centered again?
Possibly if you were in a boat. Aircraft use rudders to co-ordinate turns not to initiate or maintain them. I am not sure what you mean about the CDI, but then I suspect neither do you?
4. Let's say we were flying under NVFR, something caused the runway lights to go out, and we're running out of fuel. Would the proper procedure be to squawk 7500, inform LAX ATC we are almost out of fuel, and request an emergency ground-controlled approach landing?
Well I suppose you are near "Hollywood" and presumably this scenario is from a movie you have seen. The controller would certainly be interested on the assumption that you have also been hi-jacked!
5. Interesting question: since you can take-off, fly, land, and navigate an aircraft using only air charts and flight instruments, why does the FAA and military have a distance vision requirement? TCAS will help you avoid air traffic, so there's no real reason you should ever have to look beyond the instrument panel.
TCAS helps you avoid a collision, it isn't a navigation aid. Distance vision? Well, remember those two disasters I mentioned, and all the other comments about keeping a good lookout.
A little knowledge is a dangerous thing, and this is very little knowledge. If you are serious about aviation as a career then do as has been suggested and have a trial lesson with a view to starting the learning process towards a private licence.
Are you really 20?