I don't see any OBVIOUS problems on the vacuum hoses unless the issue is under the intake manifold.
The fact that there is NO change in how the engine runs when you remove the oil cap would indicate there is a serious vacuum leak. However, to clarify, how long has it been since your EFI fuse...
Who assembled the rings on the pistons and installed the pistons?
Sounds to me like the scraper rings did not get installed properly.
It does take time for rings to wear in and seat, but this oil consumption seems excessive.
Blow-by on the rings is what you are describing, which is...