1. No - much as they'd like to. Route would be something like KLAX SID DAG J100 LAS J146 HBU J10 OBH J100 OBK J547 FNT J94 ECK J546 MAVOD J579 YUL J509 YQY J575 YYT NATS GISTI DCT SLANY UL9 KENET STAR EGLL i.e. Jet routes in the US/Canada, then NAT across the pond, then airways to London Airport. Initial great circle true track to London Airport would be 034°, incidentally, not 340°. The practical airways/NAT/airways route is about 350 nm longer than the impractical great circle route.
2. Any worthwhile instructor will insist that you fly the aircraft with reference to visual attitude and that you lookout of the cockpit - the litany being Lookout, Attitude, Instruments. If you attempt to fly with your head glued to the panel, you will NEVER be signed off solo.
3. You won't be flying such radio navigation routes until you've learned to fly VFR. In the case you describe, the aircraft heading is adjusted by co-ordinated use of aileron and rudder so that it maintains the correct radial.
4. Under such circumstances you would divert to your alternate aerodrome rather than trying to kill yourself.
5. Very few light aircraft or military fast jets have TCAS. In any case, the requirement for distance vision is because under VFR you are responsible for 'see and avoid' collision avoidance.
An aeroplane is not a ship, things happen in three dimensions and much more quickly. You will be taught to fly using visual techniques long before you start navigating by use of radio aids.