Holley 1920 issues

As to the booster, it normal for the pedal to fall at start-up, with your foot resting on the pedal. But it should pop back up when you remove your foot. I suspect the internal valve is bad.
You can isolate the booster by clamping the vacuum line, to see how that might affect the carb.

As to the carb issue,inspect the distance from the bowl vent to the nearest surface inside the air filter house. Sometimes the carb gets sensitive if there is insufficient plenum volume, or clearance to the vent. I would also suggest to mount the filter house with no element inside it, to see if the tightening of it is somehow distorting the carb body, or affecting the linkages beneath the housing, as nm9 mentioned. Good luck