No. Only digital circuits can be implemented in FPGA. However, some FPGA are equipped with DAC and ADC, so you can use it to convert an analog signal in digital, do the amplification with digital circuits, and convert back to analog signal.
No not possible because fpga is made up of digital circuit. Fundamental building block is CLB (configurable logic block) which is made up of register, LUT, flip flop and mux. And implementation of analog amplifier not possible. Yes you can do it with extra circuitry interfaced with fpga.
Analog amplifiers need analog transistors with matching networks and bias.
FPGA can be rather used to generate the waveform and convert it from digital to analog but stays in low power levels. The use of power amplifiers is inconceivable.