damn wow, as a former microbiologist, this is painful
penicillins do work on gram negatives, but they were historically abused. The drug of choice for gonorrhea was penicillin back in the day. Syphillis is just a fancy Gram negative (genetically speaking) and structurally is organized similarly, except that it's flagella is periplasmic instead of extracellular. However, the DOC is penicillin, even though T. pallidum has an outer membrane that would act as a barrier. So nowadays, most gram negatives (and most bugs actually) are resistant to straight penicillin. That's why we have penicillin derivatives such as amp/amox/carbapenam. Vanco never penetrates the outer membrane, so it's worthless against gram negatives.
Even though Gram negatives have a thin layer of peptidoglycan, that layer is necessary for structural integrity. The thing is that with the widespread antibiotic resistance, even bugs that are sensitive will become resistant because of horizontal gene transfer. There was a paper in Nature a few years ago that showed that your gut biome has resistance genes to just about every antibiotic out there. It doesn't take much to make pathogens resistant to antibiotics, which is why we go for the heavy hitters nowadays (cipro/azithro) because even if you don't finish your course, the bugs are less likely to develop resistance because you can kill the vast majority off (>99.9%) in the few days you do take the antibiotic.
Chlyamdia has a special mechanism behind it's suspectibility to penicillin, with the best hypothesis being that it converts the EBs to RBs, allowing for immune system killing.
Mycoplasmas have no cell wall, so yea, penicillin is worthless against them.
No bacteria have a nuclear membrane. That's one of the defining characteristics between the Eukarya and Bacteria (membrane bound organelles). They have a nucleoid, which is similar, but it's not contained within a membrane.