Is MRSA and cellulitis the same?

No. Cellulitis is a skin infection. Methicillin resistant staphylococcus aureus is a type of bacteria which is one possible, and increasingly common, cause of it. It is a type of bacteria.
No, but related. Cellulitis is an skin infection that can be caused by bacteria or bacterias. Mrsa is a resistant form of a common skin bacteria. Mrsa can cause cellulitis. But, there are other bacteria that can cause cellulitis. Analogy would be house can be have insect infestation with roaches but there are other bugs that can also cause infestation.
No. Mrsa stands for methicillin-resistant staph aureus, which is a name of a specific bacteria which can cause cellulitis. Cellulitis is an infection of the skin and skin-related structures. Cellulitis can be caused by other types of bacteria as well, including streptococcus. Mrsa has been in the news more because it is now a much more common cause of cellulitis.
Completely different. Mrsa is one type of bacteria. Cellulitis is a visible inflammation (itis) of skin and soft tissues (red, hot, swollen, tender) . It can be caused by many things including sunburn (radiation), trauma (sprained ankle) or poor vein circulation (stasis dermatitis). It need not be caused by bacteria. If you have repeated episodes, get to a wound care /vascular specialist, the diagnoses you have is wron.