IMHO this is an FCC nightmare. You are likely to interfere with many radio/TV services and you won't even know it.
On to your questions.
1. L4 is to reduce the 27MHz fundamental and other lower harmonics. Without it the chances of interference with other services is greater.
2. L1 and L2 are RF chokes. Their job is to provide a low resistance DC path while providing a high impedance to the RF. Without them the circuit might not oscillate at all or oscillate at a reduced level.
3. L3 plus C3 is the primary tank circuit. Not sure how you could do without that and still have it oscillate.
4. Without a lot of thought I might agree with you on the transistor type, but you do need some good gain at 27MHz.
5. The diodes DO need to be high frequency, low capacitance types. They are responsible for clipping the output wave to produce the harmonics. If they are not high frequency types then the wave will still be clipped but will not contain as many high frequencies as the clipped edges will be softer and therefore contain less harmonics.
Since it's a brute force type of RF generator any of the modifications you suggest might still allow it to work but probably not as well. But since there is no specification then who is to say how well it needs to work.
Ray