Stack frame is the packed information related to a function call. This information generally includes arguments passed to th function, local variables and where to return upon terminating. Activation record is another name for a stack frame. The layout of the stack frame is determined in the ABI by the manufacturer and every compiler supporting the ISA must conform to this standard, however layout scheme can be compiler dependent. Generally stack frame size is not limited but there is a concept called "red/protected zone" to allow system calls...etc to execute without interfering with a stack frame.
There is always a SP but on some ABIs (ARM's and PowerPC's for example) FP is optional. Arguments that needed to be placed onto the stack can be offsetted using the SP only. Whether a stack frame is generated for a function call or not depends on the type and number of arguments, local variables and how local variables are accessed generally. On most ISAs, first, registers are used and if there are more arguments than registers dedicated to pass arguments these are placed onto the stack (For example x86 ABI has 6 registers to pass integer arguments). Hence, sometimes, some functions do not need a stack frame to be placed on the stack, just the return address is pushed onto the stack.